Vibrations of Thick Cylindrical Structures
Hamid R. Hamidzadeh · Reza N. Jazar
Vibrations of Thick Cylindrical Structures
123
Hamid R. Hamidzadeh Department of Mechanical Engineering Tennessee State University 3500 John A. Merritt Blvd. Nashville TN 37209-1561 USA
[email protected] Reza N. Jazar Department of Mechanical and Automotive Engineering School of Aerospace, Mechanical and Manufacturing Engineering RMIT University Melbourne, VIC 3083 Australia
[email protected] ISBN 978-0-387-75590-8 e-ISBN 978-0-387-75591-5 DOI 10.1007/978-0-387-75591-5 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2009937454 c Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Contents 1 Introduction 1.1 Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Vibrations of Cylindrical Structures . . . . . . . . . . . . 1.3 Vibrations of Thin Shell Cylinders . . . . . . . . . . . . . 1.4 Vibrations of Rods . . . . . . . . . . . . . . . . . . . . . . 1.5 Vibration of Thick Cylindrical Structures . . . . . . . . . 1.6 Vibrations of Damped Cylindrical Structures . . . . . . . 1.7 Vibration of Multilayered Cylindrical Structures . . . . . 1.8 Vibrations of Constrained Layered Cylindrical Structures 1.9 Vibrations of Cylindrical Panels . . . . . . . . . . . . . . .
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1 1 2 3 4 5 7 8 10 12
2 Governing Equations 2.1 State of Stresses at a Point . . . . . . . . 2.2 Equilibrium Equations in Terms of Stress 2.3 Stress-Strains Relationships . . . . . . . . 2.4 Strain-Displacement Relationships . . . . 2.5 Stress-Displacement Relationships . . . . 2.6 Equations of Motion . . . . . . . . . . . . 2.7 Key Symbols . . . . . . . . . . . . . . . .
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3 Vibration of Single-Layer Cylinder 3.1 Governing Equation . . . . . . . . . . . . . . . . . . . . . . 3.2 Proposed Solution . . . . . . . . . . . . . . . . . . . . . . . 3.3 Derivation of Wave Equation . . . . . . . . . . . . . . . . . 3.4 Solutions to the Wave Equations . . . . . . . . . . . . . . . 3.4.1 Derivation of Bessel Equation Bn,αx [f ] = 0 . . . . . 3.4.2 Derivation of the Bessel Equation Bn,βr [gz ] = 0 . . . 3.4.3 Derivation of the Bessel Equations Bn+1,βr [gr − gθ ] = 0 and Bn−1,βr [gr + gθ ] = 0 . . . . . . . . . . . . . . 3.5 Solutions to the Bessel differential equations . . . . . . . . . 3.6 Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Derivation of the Radial Stress σrr . . . . . . . . . . 3.7.2 Derivation of the Shear Stress τ rθ . . . . . . . . . . 3.7.3 Derivation of the Shear Stress τ rz . . . . . . . . . . 3.8 The Boundary Stresses . . . . . . . . . . . . . . . . . . . . . 3.9 The Stress-Displacement Matrix Equation . . . . . . . . . . 3.10 Governing Equations for Free and Forced Vibrations . . . .
29 30 30 31 32 33 34 34 36 36 37 38 39 39 40 42 43
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3.11 Key Symbols . . . . . . . . . . . . . . . . . . . . . . . . . .
45
4 Modal Analysis of Cylindrical Structures 4.1 Frequency Factor . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Geometry of the Cylinder . . . . . . . . . . . . . . . . . . . 4.3 Properties of the Medium . . . . . . . . . . . . . . . . . . . 4.4 Axial Wavelength . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Modes of Vibration . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Modes with Different Circumferential Wave Number 4.5.2 Thickness Modes . . . . . . . . . . . . . . . . . . . . 4.5.3 Modes with Infinite Axial Wavelengths . . . . . . . . 4.5.4 Modes with Finite Axial Wavelengths . . . . . . . . 4.5.5 Axially Symmetric Modes . . . . . . . . . . . . . . . 4.6 Forced and Free Modal Analysis . . . . . . . . . . . . . . . 4.7 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Key Symbols . . . . . . . . . . . . . . . . . . . . . . . . . .
47 47 47 48 48 49 50 52 54 54 55 56 77 78
5 Vibration of Multi-Layer thick cylinders 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Historical Background . . . . . . . . . . . . . . . . . . 5.3 Scope of the Chapter . . . . . . . . . . . . . . . . . . . 5.4 Modal Displacements and Stresses . . . . . . . . . . . 5.5 Propagator Matrix for a Three Layer Cylinder . . . . 5.6 Propagator Matrix for a Multi-Layered Thick Cylinder 5.7 Natural Frequencies and Modal Loss-Factors . . . . . 5.8 Sample Calculations for One Layered Cylinder . . . . 5.9 Modal Identification . . . . . . . . . . . . . . . . . . . 5.9.1 Axially Symmetric and Asymmetric Vibrations 5.9.2 Thickness Modes . . . . . . . . . . . . . . . . . 5.10 Case Studies . . . . . . . . . . . . . . . . . . . . . . . 5.10.1 Elastic Cylinder . . . . . . . . . . . . . . . . . 5.10.2 Two-Layer Elastic Cylinder . . . . . . . . . . . 5.10.3 Three-layer Elastic Cylinder . . . . . . . . . . . 5.11 Key Symbols . . . . . . . . . . . . . . . . . . . . . . .
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79 . 79 . 80 . 81 . 82 . 83 . 86 . 87 . 88 . 90 . 91 . 91 . 95 . 95 . 97 . 99 . 101
6 Constrained Layer Damping Treatment of Thick Cylindrical Structures 103 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.2 Scope of the Chapter . . . . . . . . . . . . . . . . . . . . . . 105 6.3 Natural Frequencies and Modal Loss-Factors . . . . . . . . 105 6.4 The Effects of the Core Loss-Factors on Modal Parameters 107 6.5 The Effects of Various Shear Moduli for the Middle Layer . 119 6.6 The Effects of Various Thicknesses for The Middle Layer . . 129 6.7 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6.8 Key Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Contents
7 Vibrations of Partial Cylindrical Panels 7.1 Introduction . . . . . . . . . . . . . . . . . . 7.2 Objectives of the Present Chapter . . . . . 7.3 Harmonic Vibrations of Cylindrical Panels . 7.3.1 Modified Equations of Motion . . . . 7.3.2 Solution to the Modified Equation of 7.3.3 Modal Displacements . . . . . . . . 7.3.4 Modal Stresses . . . . . . . . . . . . 7.4 Modal Displacement and Stresses . . . . . . 7.5 Modal Analysis . . . . . . . . . . . . . . . . 7.6 Computed Resonant Frequencies . . . . . . 7.7 Finite Element Model . . . . . . . . . . . . 7.8 Concluding Remarks . . . . . . . . . . . . . 7.9 Key Symbols . . . . . . . . . . . . . . . . .
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153 153 154 155 155 157 158 159 160 161 163 184 186 187
References
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A Trigonometric Formulae
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Index
199
List of Figures 2.1 2.2 2.3
Direct Stresses in Cylindrical Coordinates. . . . . . . . . . . Stresses in the r and θ Directions. . . . . . . . . . . . . . . Stresses in the plane perpendicular to r and z direction. . .
15 16 17
2.4 2.5 2.6 2.7
An Element in Cylindrical Coordinates. . Element Subjected to Small Deformation. Horizontal Plane of Strained Element. . . The (z, θ) Plane of Strained Element. . .
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3.1
Reference Coordinates and Dimensions. . . . . . . . . . . .
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4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14
Fundamental Axial Mode. . . . . . . . . . . . . . . . . Second Axial Mode. . . . . . . . . . . . . . . . . . . . Breathing Mode n = 0. . . . . . . . . . . . . . . . . . . Bending and Axial Shear Modes n = 1. . . . . . . . . First Lobar Mode, n = 2. . . . . . . . . . . . . . . . . Second Lobar Mode, n = 3. . . . . . . . . . . . . . . . Third Lobar Mode, n = 4. . . . . . . . . . . . . . . . . Fourth Lobar Mode, n = 5. . . . . . . . . . . . . . . . Rigid Body Flexural Mode, n = 1. . . . . . . . . . . . Modes Associated with (n = 0) and One Node Circle. Modes Associated with (n = 0) and Two Node Circle. Radial Axisymmetric Mode. . . . . . . . . . . . . . . . Axisymmetric Longitudinal and Breathing Modes. . . Axisymmetric Torsional Mode. . . . . . . . . . . . . .
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5.1 5.2 5.3 5.4
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Cross Section of a Three Layered Cylinder. . . . . . . . . . Cross Section of a Multi-Layered Circular Cylinder. . . . . The geometrical properties of a cylinder. . . . . . . . . . . . Torsional Mode and the First Three Related Thickness Modes, n = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Radial Mode and the First Three Thickness Modes, n = 0. . 5.6 Longitudinal Mode and the First Three Thickness Modes, n = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Bending mode for n = 1. . . . . . . . . . . . . . . . . . . . . 5.8 Breathing Mode, n = 0. . . . . . . . . . . . . . . . . . . . . 5.9 Rigid Body Mode, n = 1. . . . . . . . . . . . . . . . . . . . 5.10 First Lobar Mode, n = 2. . . . . . . . . . . . . . . . . . . .
83 87 89 91 92 92 92 93 93 94
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List of Figures
5.11 Second Lobar Mode, n = 3. . . . . . . . . . . . . . . . . . . 5.12 Third Lobar Mode, n = 4. . . . . . . . . . . . . . . . . . . . 6.1 6.2
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6.12 6.13 6.14 6.15 6.16
A typical three-layered sandwich cylindrical structure with visco-elastic middle layer. . . . . . . . . . . . . . . . . . . . Variation of the natural frequency factors for the first three modes versus the material loss-factor of the viscoelastic layer for n = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variation of the natural frequency factors for the first three modes versus the material loss-factor of the viscoelastic layer for n = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variation of the natural frequency factors for the first three modes versus the material loss-factor of the viscoelastic layer for n = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variation of the natural frequency factors for the first three modes versus the material loss-factor of the viscoelastic layer for n = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variation of the natural frequency factors for the first three modes versus the material loss-factor of the viscoelastic layer for n = 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variation of the modal loss factors for the first three modes versus the material loss-factor of the viscoelastic layer for n = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variation of the modal loss factors for the first three modes versus the material loss-factor of the viscoelastic layer for n = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variation of the modal loss factors for the first three modes versus the material loss-factor of the viscoelastic layer for n = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variation of the modal loss factors for the first three modes versus the material loss-factor of the viscoelastic layer for n = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variation of the modal loss-factors for the first three modes versus the material loss-factor of the viscoelastic layer for n = 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variation of natural frequency factors for the first three modes versus ratio of G02 /G1 for n = 0. . . . . . . . . . . . . . . . . Variation of natural frequency factors for the first three modes versus ratio of G02 /G1 for n = 1. . . . . . . . . . . . . . . . . Variation of natural frequency factors for the first three modes versus ratio of G02 /G1 for n = 2. . . . . . . . . . . . . . . . . Variation of natural frequency factors for the first three modes versus ratio of G02 /G1 for n = 3. . . . . . . . . . . . . . . . . Variation of natural frequency factors for the first three modes versus ratio of G02 /G1 for n = 4. . . . . . . . . . . . . . . . .
94 94 106
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118 130 130 131 131 132
List of Figures
6.17 Variation of modal loss factors for the first three modes versus ratio of G02 /G1 for n = 0. . . . . . . . . . . . . . . . . . 6.18 Variation of modal loss factors for the first three modes versus ratio of G02 /G1 for n = 1. . . . . . . . . . . . . . . . . . 6.19 Variation of modal loss factors for the first three modes versus ratio of G02 /G1 for n = 2. . . . . . . . . . . . . . . . . . 6.20 Variation of modal loss factors for the first three modes versus ratio of G02 /G1 for n = 3. . . . . . . . . . . . . . . . . . 6.21 Variation of modal loss factors for the first three modes versus ratio of G02 /G1 for n = 4. . . . . . . . . . . . . . . . . . 6.22 Variation of natural frequency factors for the first three modes versus thickness of viscoelastic layer for n = 0. . . . . . . . 6.23 Variation of natural frequency factors for the first three modes versus thickness of viscoelastic layer for n = 1. . . . . . . . 6.24 Variation of natural frequency factors for the first three modes versus thickness of viscoelastic layer for n = 2. . . . . . . . 6.25 Variation of natural frequency factors for the first three modes versus thickness of viscoelastic layer for n = 3. . . . . . . . 6.26 Variation of natural frequency factors for the first three modes versus thickness of viscoelastic layer for n = 4. . . . . . . . 6.27 Variation of modal loss factors for the first three modes versus ratio of G02 /G1 for n = 0. . . . . . . . . . . . . . . . . . 6.28 Variation of modal loss factors for the first three modes versus ratio of G02 /G1 for n = 1. . . . . . . . . . . . . . . . . . 6.29 Variation of modal loss factors for the first three modes versus ratio of G02 /G1 for n = 2. . . . . . . . . . . . . . . . . . 6.30 Variation of modal loss factors for the first three modes versus ratio of G02 /G1 for n = 3. . . . . . . . . . . . . . . . . . 6.31 Variation of modal loss factors for the first three modes versus ratio of G02 /G1 for n = 4. . . . . . . . . . . . . . . . . . 7.1 7.2 7.3 7.4 7.5 7.6 7.7
Coordinates and displacements of a typical thick visco-elastic cylindrical panel. . . . . . . . . . . . . . . . . . . . . . . . . Modal displacement patterns for thick panel for n = 0 and n = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modal displacement patterns for a thick panel when n = 2 and n = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modal displacement patterns for thick panel for n = 4 and n = 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The variation of the first five resonant frequency factors versus panel angle for n = 1, H/R = 0.3, η = 0.0. . . . . . . . . The variation of the first five resonant frequency factors versus panel angle for n = 2, H/R = 0.3, η = 0.0. . . . . . . . . The variation of the first five resonant frequency factors versus panel angle for n = 3, H/R = 0.3, η = 0.0. . . . . . . . .
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132 133 133 134 134 146 146 147 147 148 148 149 149 150 150 155 162 162 163 167 167 168
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List of Figures
7.8 7.9 7.10 7.11 7.12 7.13 7.14 7.15 7.16
The variation of the first five resonant frequency factors versus panel angle for n = 4, H/R = 0.3, η = 0.0. . . . . . . . . The variation of the fundamental frequency factors versus the ratio H/R for n = 1, 2, 3, 4, and panel angle of 40 deg. The variation of the fundamental frequency factors versus the ratio H/R for n = 1, 2, 3, 4, and panel angle of 60 deg. The variation of the fundamental frequency factors versus the ratio H/R for n = 1, 2, 3, 4, and panel angle of 80 deg. The variation of the fundamental frequency factors versus the ratio H/R for n = 1, 2, 3, 4, and panel angle of 100 deg. The variation of the fundamental frequency factors versus the ratio H/R for n = 1, 2, 3, 4, and panel angle of 120 deg. The variation of the fundamental frequency factors versus the ratio H/R for n = 1, 2, 3, 4, and panel angle of 140 deg. The variation of the fundamental frequency factors versus the ratio H/R for n = 1, 2, 3, 4, and panel angle of 160 deg. The variation of the fundamental frequency factors versus the ratio H/R for n = 1, 2, 3, 4, and panel angle of 180 deg.
168 172 172 173 173 174 174 175 175
Dedicated to our wives Azar and Mojgan.
Happiness is thinking by yourself, not by someone or something else, not for someone or something else.
Preface
The authors have endeavored to develop a book that is the result of over three decades of academic careers, conducting research on various aspects of Vibration Engineering, and dissemination of their research results. The book introduces the reader to the analytical techniques for vibration analysis and damping of thick cylindrical structures where linear elastic shell theories cannot offer realistic solutions in this important engineering problems. This book presents necessary theoretical background for design analysis. Furthermore, it offers the engineering information and quantitative data needed for design analysis, applications, and construction of these structures. The book, in its entirety, constitutes an extensive guide for the reader. It also provides a systematic solution for the modal analysis of thick cylindrical structures and panels, and extends the solution to vibration of layered cylindrical structures. The book is intended to lay the foundation for understanding mathematical modeling, vibration analysis, damping, and the design of thick elastic and visco-elastic single-layer and multi-layer cylindrical structures in a complete and succinct manner. Throughout the book, an attempt has been made to provide a conceptual framework that includes exposure to the required background in mathematics, fundamentals of the theory of elasticity, modal analysis, and constrained-layer damping treatment, as well as Bessel differential equations and their solutions. The knowledge of the presented topics will enable the reader to pursue further advances in the field. Level of the Book The primary audience of this book is the graduate students in mechanical engineering, engineering mechanics, civil engineering, aerospace engineering, ocean engineering, mathematics, and science disciplines. In particular, it is geared toward students interested in enhancing their knowledge by taking the second graduate course in the areas of vibration of continuous systems, structural dynamics, and application of passive damping treatments. The presented topics have been prepared to serve as an aid to designers of cylindrical structures. It can also be utilized as a guide to practicing engineers who are responsible for setting design specifications and ensuring
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their fulfillment. The covered topics are also of interest to research engineers who are seeking to expand their expertise in these areas. Organization of the Book The book is presented in seven chapters: introduction, fundamentals of elasticity, vibration analysis for single-layer cylinders, modal analysis for single-layer cylinders, vibration of multi-layer thick cylinders, constrainedlayer damping for cylindrical structures, and vibration of thick cylindrical panels. Furthermore, it offers helpful and significant tabulated results, which can be used as design guidelines for these structures. To make effective use of the presented topics, the following procedure is suggested. A working understanding of certain theoretical concepts and method can be achieved by referring to chapters 1 through 3. To become acquainted with the state of the art in this particular field and learn about the historical background on this topic, the reader should begin with chapter 1, which contains an extensive list of key references with brief discussions on their methodologies, required assumptions, and their achievements. Chapter 2 reviews the succinct fundamental theoretical background and concepts needed from the theory of elasto-dynamics, which will enable the reader to follow the derivation of the required governing equations and their solutions in chapter 3. Chapter 4 is intended to present numerical results for the natural frequencies and mode shapes of cylindrical structures with a variety of geometries. Chapters 5 and 6 present analytical modeling, and informative case studies for vibration analysis of the multilayer cylindrical structures and the constrained layer damping treatment of these structures. Chapter 7 is designed to provide the essential modal analysis for long, thick, cylindrical panels based on the plane strain assumption. Method of Presentation The scope of each chapter is clearly outlined and the governing equations are derived with an adequate explanation of the procedures. The covered topics are logically and completely presented without unnecessary overemphasis. The topics are presented in a book form rather than in the style of a handbook. Tables, charts, equations, and references are used in abundance. Proofs and derivations are often emphasized, and the physical model and final results are accompanied with illustrations and interpretations. Specific information that is required in carrying out the design analysis in detail has been stressed. Prerequisites This book is primarily intended for graduate students, so the assumption is that the readers are familiar with the fundamentals of vibration engineering and differential equations, as well as a basic knowledge of linear algebra and numerical methods. The presented topics are given in a
Preface
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way to establish a conceptual framework that enables the reader to pursue further advances in the field. Although the governing equations will be derived with adequate explanations of the procedures, it is assumed that the readers have a working knowledge of mechanical vibration and the theory of elasticity. Unit System To maintain generality, computed results and the required parameters are provided in non-dimensional forms. Nevertheless, the system of units adopted for case studies is, unless otherwise stated, the British Gravitational system of units (BG). The units of degree (deg) or radian ( rad) are utilized for variables representing angular quantities.
Acknowledgements We wish to express our thanks to our colleagues who have assisted in the development of this book. We are indebted to our friends and colleagues Hurang Hu, Albert C. J. Luo, and Annie Tangpong for their constructive comments and suggestions.
1 Introduction Due to strong potential applications and more demanding requirements imposed upon applications of cylindrical structures, there has been increasing research and development activities during recent years in the field of vibration analysis and damping treatment for this type of structure. An important step in vibration analysis of cylindrical structures is the evaluation of their vibration modal characteristics, such as natural frequencies, mode shapes, and modal loss-factors. This modal information plays a key role in the design and vibration suppression of these structures when subjected to dynamic excitations. Most reported studies on the dynamic response of cylindrical structures have been restricted to the application of shell theories. These theories are based on a number of simplifying assumptions, the most important of which is that the considered shell must be relatively thin to assume constant stresses within the cylinder. Therefore, due to this limitation, shell theories are inadequate to accurately describe many modes of vibrations which occur in thick cylindrical structures. The primary scope of this chapter is to address a brief historical background on the key reported investigations on this topic. It should be noted that there is extensive literature available on vibration analysis and damping treatments of thin cylindrical shells and panels. However, due to the main scope of this book, this chapter is more concerned with reporting the research results for thick cylindrical structures. Nevertheless, some major reported developments addressing the vibration analysis and damping treatments for thin cylindrical structures are also presented.
1.1 Vibrations The phenomenon of vibration was used by ancient peoples in attempts to produce music. One of the first widely recognized mathematical studies of vibration was Galileo’s study of pendulums and the transfer of energy between two bodies tuned to the same natural frequency (Timoshenko, 1953). Galileo continued his research into the vibration of strings. Along with Hooke, Galileo showed the relationship between sound and the frequency of mechanical vibration (Timoshenko, 1953). Many mathematicians H.R. Hamidzadeh, R.N. Jazar, Vibrations of Thick Cylindrical Structures DOI 10.1007/978-0-387-75591-5_1, © Springer Science+Business Media, LLC 2010
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1. Introduction
contributed to the study of vibrations. Some of the prominent mathematicians include Taylor, Bernoulli, D’Alembert, Euler, Lagrange, and Fourier (Rao, 1990). Wallis and Sauveur independently observed the mode shape and node formation phenomenon in the vibration of strings (Cannon and Dostrovsky 1981). Bernoulli was credited with the principle of linear superposition of harmonics (Cannon and Dostrovsky 1981). In the seventeenth century, Hooke presented his law of elasticity. Euler and Bernoulli used Hooke’s law to produce the governing differential equations for the lateral vibration of prismatic bars, and solved the equations for small displacements, (Rao, 1990). Coulomb expanded upon their research by investigating the torsional vibration of wire and presenting the results. One of Lagrange’s many contributions was in correcting the work of Sophie Germain in 1809. This research mathematically described the theory of plate vibration. Lagrange’s contribution was later corrected by Kirchhoff in 1850 (Bucciarelli and Dworsky, 1980). In 1877, Lord Rayleigh published his book on the theory of sound. He later produced a method for finding natural frequencies by the principle of conservation of energy (Lindsay, 1970). Rayleigh’s work is still significant today. Vibration research is also described as the study of wave propagation. One of the most historically important studies of wave propagation was produced by Horace Lamb (1904). Lamb studied the propagation of vibrations over the surface of a “semi-infinite” isotropic elastic solid, more conventionally known as an elastic half space, subjected to an impulse load at an arbitrary point on the surface. Modern contributors to the theory of vibration are Timoshenko, Stodola, and Mindlin. Stodola added to the topic of turbine blade vibration. Timoshenko and Mindlin improved theories about the vibration of beams and plates. Most modern textbooks that introduce mechanical vibration contain many direct references to Timoshenko’s work. For the interested reader, textbooks written by Timoshenko (1953) contain many of the same theories that are currently in use.
1.2 Vibrations of Cylindrical Structures Cylindrical structures are widely used in the construction of structures from the fuselage of a jet airplane or missile to pressure and piping vessels. A greater understanding of these structural components is desired to prevent failure and to optimize their design. This book is intended to present vibration analysis and the modal characteristics of these structures. The
1. Introduction
3
identification of the resonant frequency of these structures and the effect of various levels of material damping on the response are intended to help in the design of these systems. Thus, the effect of different elastic and viscoelastic materials on the vibration of a typical thick cylinder is considered. This book begins with descriptions of the technique used in the study of the forced vibration of visco-elastic cylinders. It then concerns with the study of the relation between the material loss-factor and the modal damping loss-factors associated with different mode shapes.
1.3 Vibrations of Thin Shell Cylinders Due to the complexity of using the theory of elasticity for all cylindrical structures, the theory of shells was developed to address dynamic analysis of thin cylindrical structures. Extensive research is reported for different aspects of vibration, stability and control of the cylindrical structures. In the majority of the reported research, it was common to use shell theories. However, the theory of shells makes many approximations. It should be noted that the shell considered must be relatively thin to justify constant stresses within the thickness of the shell. Several techniques have been used to study the vibrations of hollow cylinders. Initially, the simplest technique, the shell theory, was used. Rayleigh (1894) determined the natural frequencies of thin cylinders with free ends while considering axial vibration. Love (1927) studied the flexural vibration of cylinders. Flugge (1934), by a similar approach, succeeded in obtaining a frequency equation for a cylinder with freely supported ends. The three roots of his equation defined three natural frequencies for any given nodal pattern. The problem was further investigated by Arnold and Warburton (1949). They attempted to explain certain frequency phenomena that were observed in experiments on thin cylinders. For a cylinder with freely supported ends, they derived the frequency equations and were able to verify the experimental results with considerable accuracy. They found that for short cylinders with very thin walls, the natural frequencies may actually decrease as the number of circumferential nodes increases. This was shown theoretically to be due to the proportion of strain energy contributed respectively by bending and stretching. Arnold and Warburton (1953) considered the flexural vibrations of thin elastic cylinders. In this type of vibration, many forms of nodal patterns may exist due to the combination of circumferential and axial nodes. Theoretical expressions were developed for the natural frequencies of cylinders
4
1. Introduction
with freely-supported and fixed ends. They reported an acceptable agreement between the theoretical and the experimental natural frequencies. Forsberg (1964) implemented Flugge’s equations of motion to determine natural frequencies for uniform thin shells, regardless of the end conditions. His method assumed a natural frequency and sought a length of cylinder, this was different than all other methods previously used. Warburton (1965), using Forsberg’s method, investigated thin shells with clamped and free ends. Warburton’s concluded that for long wavelengths and large numbers of circumferential waves the effect of the end conditions was negligible. Therefore, it was possible to treat the ends as simply supported for frequency calculations. An account of the various thin shell methods can be found in the introductory chapter of Thin-Shell Structures: Theory, Experiment, and Design, edited by Fung and Sechier (1972). Extensive discussions on the applications of thin shell theories are provided in a NASA Technical Report, by Leissa (1973), and also in Pietraszkiewiz and Szymcvak (2006), and Amabili (2008). Needless to say, the shell theories are inadequate to accurately describe many modes of vibration in shells, as well as thick cylinders. Therefore, due to the limitations associated with the shell theory, the elasto-dynamic theory should be employed.
1.4 Vibrations of Rods As researchers gained interest in the dynamics of cylinders, the vibration analysis of solid shafts was considered. Pochhammer (1876) was the first to study the vibrations of solid cylinders. Chree (1889) studied infinitely long circular rods with traction-free circumferential boundaries. Both investigators utilized the theory of elasticity in their research. Using the theory of elasticity, McNiven and Perry (1962) discussed the coupling of longitudinal axial shear and radial modes in an infinitely long rod. McNiven et al. (1966) proposed a "three-modes theory" for rods that compared well with elasto-dynamic theory. In this study, the first three modes of vibration are assumed to be dominant, while the other modes are assumed to contribute little to the overall response. Rumerman and Raynor (1971) studied the radial and axial modes of an infinitely long cylinder using the Rayleigh-Ritz method. Since structural cylinders are not of infinite length, the next logical step was to investigate rods of finite length rods. Hutchinson (1972) studied axisymmetric free vibrations of finite length rods using a series solution to the three-dimensional theory of elasticity. McMahon (1964) published his experimental results for
1. Introduction
5
natural frequencies of solid cylinders with free boundaries. These important experimental results are not adequate for determination of modes and geometric or elastic parameters. Rasband (1975) extended Hutchinson’s earlier work to study non-axisymmetric free vibrations in a solid cylinder of finite length. Hutchinson (1980) then extended his work to include nonaxisymmetric vibrations of a solid rod and compared his results to those obtained experimentally by McMahon (1964). Lusher and Hardy (1988) studied axisymmetric free vibration of a transversely isotropic finite cylindrical rod.
1.5 Vibration of Thick Cylindrical Structures The shell theory may be applied to many thin cylindrical structures, while thicker cylinders can be accurately studied only by using the three-dimensional theory of elasticity. It should be noted that the theory of elasticity can be applied to thick or thin cylindrical structures. Among the earlier thorough studies of infinitely long traction-free hollow cylinders, using the theory of three dimensional elasticity, one can mention the independent works reported by Greenspon (1957) and Gazis (1959). Gazis (1959) studied the free harmonic wave propagation along a hollow circular cylinder of infinite extent, within the framework of the linear theory of elasticity. He derived a characteristic equation appropriate to the circular hollow cylinder by using the Helntholtz potential for arbitrary values of the physical parameters involved. Armenakas (1967) derived the frequency equation for harmonic waves, with an arbitrary number of circumferential nodes, traveling in traction free composite, circular cylindrical structures. Armenakas et al. (1969) considered the transmission of elastic energy by means of elastic waves, and formulated the eigenvalue problem for stress-free cylinders for different circumferential wave numbers. They also presented tables of non-dimensional natural frequencies for a wide range of geometrical possibilities. Their analysis was based on the linear threedimensional theory of elasticity. In a separate article, Armenakas (1971) provided numerical results for evaluating the effect of changes of the shell parameters on the frequency and shape of the first few modes of vibrations. The composite shells, used in Armenakas et al. (1969), consisted of infinitely long cylinders with two concentric circular shells bounded at the interfaces. Hutchinson (1979) studied the axisymmetric flexural vibrations of thick free circular plates by using the general three-dimensional linear elasticity
6
1. Introduction
to find accurate natural frequencies and mode shapes for the flexural vibrations of free circular plates. He included shear and rotary inertia effects, compared them with the accurate series solution used by Pickett (1944), and found that the approximate solution yielded frequencies with sufficient accuracy for most engineering applications. Hutchinson (1980) also applied the series solution method to find the natural frequencies of solid vibrating cylindrical rods. Hutchinson and Ei-Azhari (1986) extended Hutchinson’s work in solid cylinders to include free hollow cylinders with finite length. Other approaches for the vibration analysis of hollow cylindrical structures have been proposed. McNiven et al. (1966) used a “three-mode theory” to obtain good agreement with the exact solution based on the threedimensional theory of elasticity. The three-mode theory is based on the assumption that the first three (lowest) modes have a high influence on each other and the other modes are less significant. Gladwell and Vijay (1975) used a finite element approach to the three-dimensional vibration of finite length circular cylinders with traction-free surfaces and free-end conditions. Svardah (1984) treated wave propagation in semi-infinite, elastic cylinders with traction-free lateral surfaces initially at rest and excited by transient end loading. Markus (1988) studied the effect of visco-elastic material on the vibration of cylindrical structures using the shell theory. Singal and Williams (1988) developed approximate solutions for natural frequencies of thick cylinders and rings, and verified their results by conducting experiments. Stephen and Wang (1992) employed the Papleovitch-Nueber solution for the elasto-static displacement equations of equilibrium, and derived both axial and non-axisymmetric solutions. They considered zero traction on the surface generators of the infinitely long elastic cylinder and formulated an eigen equation for free vibration. They computed smaller roots of the eigen equation for the circumferential harmonic numbers of n = 0, 1, 2 and 3 for different wall thickness ratios. A formal mathematical review of elasto-dynamic problems, such as the one presented here, is presented by Mal and Singh (1991). The free vibration solution to hollow cylindrical dynamics prompted the next step of research which was to determine the forced response of the cylinder. Other interesting work in the area of forced vibrations can be found in research work reported by Braga et al. (1990). Braga’s research consisted of the study of wave propagation in a fluid-loaded laminated cylinder The effects of the laminates and cylinder curvature on the dispersion curves of the subsonic interface waves were given particular attention. In this study, the three-dimensional elastic theory was used for accurate analysis of acoustical waves. This investigation is of interest due to the similar procedure for choosing appropriate Bessel function values for the
1. Introduction
7
solution to the elasticity problem. Several forced vibration solutions have also been reported using the shell theory. When using this theory, many dynamic effects have been neglected in order to obtain a closed form solution. An example of this is the impulse loading of orthotropic cylinders by Christoforou and Swanson (1990) using the shell theory. Many forced vibration problems have not yet been solved using the linear elastic theory. Hamidzadeh et al. (1991) studied the forced vibrations of hollow cylinders. They employed the three-dimensional elastic theory and developed the governing equations of motion in terms of volumetric strain and elastic rotations. They then produced a system of linear non-homogeneous equations and solved for the stresses and displacements for a wide range of excitation frequencies. Using a frequency sweeping procedure, they calculated resonant frequencies and compared their results to those of Gazis (1959) and found dissimilar results for short cylinders and fundamental resonant modes. Hamidzadeh et al. (1991), and Hamidzadeh and Moxey (2004) also studied the harmonic responses of visco-elastic thick circular cylinders of infinite extent. The cylinders were subjected to harmonic radial and tangential boundary stresses. The two-dimensional theory of elasticity and a complex elastic modulus were used to provide a solution for stresses and displacements at any point in the cylinder. Hamidzadeh and Chandler (1991) extended their research work to develop a two-dimensional visco-elastic theory involving multi-layered sandwich cylinders. The solution to the three-layer cylinders was developed by utilizing the stresses and displacements for the single cylinder and by complying with the compatibility requirements at each interface. Their computed resonant frequency for a three-layer elastic cylinder with the same properties compared satisfactorily with those computed for a single-layer cylinder having identical geometry.
1.6 Vibrations of Damped Cylindrical Structures The main objective in using visco-elastic material is to dissipate energy and reduce the amplitude of the vibration. The visco-elastic material can be mathematically modeled using complex stiffness and moduli. This is due to the phase difference between stress and strain. Visco-elastic materials are very resilient and are widely used for damping treatments. The use of such material reduces the amplitude of stresses and displacements for a longer working life. This section contains an overview of the past research works on the effect of material damping on vibration characteristics of thick cylindrical structures. A notable work was conducted by Parfitt and
8
1. Introduction
Lambeth, (1962) who studied the damping and transmission of waves in visco-elastic material structures. Warburton (1965) showed the response for shells under loading, for different material loss-factors. While most research for determination of the modal analysis area is focused on finding the actual resonant frequencies of the cylindrical structures, few have sought to find the actual effect of the differing geometric and material properties on the vibration of the cylinders. Lazan (1968) uses the definition of rheology to include elastic deformation, plastic creep flow, creep relaxation, and damping. Flugge (1967) and Snowdon (1968) have shown the process of considering “rubber-like materials” as having a complex elastic modulus by considering the relationship of stress to strain with linear partial derivatives. Ewins (1986) has given a mathematical method for determining the modal loss-factor of structures based on circle-fitting the response data in the vicinity of the resonance. Yeh and Yang (1987) have described a time domain technique for the modal identification of a system. Mead and Markus (1970) determined modal frequencies and loss-factors for an incasters damping sandwich beam. Crandall (1991) has described damping models in the time domain for the ideal viscous damper, the ideal hysteretic damper, and the band-limited hysteretic damper. Pang et al. (1992) and Bergman (1993) considered the effect of visco-elastic materials on vibration control of actuators and hybrid control of Euler-Bernoulli beams. Hamidzadeh and Chandler (1991) applied the general three-dimensional theory of elasticity to study the harmonic forced vibration of thick viscoelastic hollow cylinders. Resonant frequencies of the breathing and first three lobar modes for cylinders were computed. Their results compared well with those of Armenakas et al (1969).
1.7 Vibration of Multilayered Cylindrical Structures Composite multilayer cylinders have widespread use in engineering. Pipes, aircraft, submarines, missiles, rockets, and power transmission shafts are typical cylindrical structures. When the frequency of an acting force on a cylinder coincides with the natural frequency of the structure, the phenomenon of resonance occurs. Resonance is associated with large displacements and stresses and may cause failure in the cylinder. Therefore, determining the natural frequency is vital and has a direct impact on the design of composite cylindrical structures.
1. Introduction
9
Many researchers have investigated the vibration of cylinders using different analytical approaches, such as: the thin shell theory, the curved beam theory, and the theory of elasticity. Most studies on the vibrations of constrained layer damped cylinders are based on the assumption of thin shells. The assumptions used in the shell theory for cylindrical structures may not always be valid. In fact, a significant number of cylindrical structures are considered to be thick. In spite of this, few studies have been directed to the analysis of the free vibrations of thick cylindrical structures. Crampin (1970) studied desperation of surface waves in multi-layered and isotropic media. Bhaskar and Varadan (1990) developed a solution for analysis of pinched laminated cylinders. Rattanawanqcharoen and Shah (1990) studied the wave propagation in a laminated isotropic cylinder. The dispersion relation for a multi-layered elastic cylinder was evaluated by a propagator matrix and nondimensional natural frequencies for a twolayered cylinder with a special circumferential wave number, n = 1, and a range of axial wave numbers were calculated. Hawkes and Soldatos (1992) studied the vibrations of multi-layered laminated cylinders. They achieved the solution using a method of successive approximations. The exact governing equations of the cylinders were replaced by a set of simpler approximate equations that were solved analytically. They compared their results with those of Armenakas et al. (1969). Huang and Dong (1984) utilized a stiffness method to consider wave propagation in laminated anisotropic cylinders with an arbitrary number of lamina. Braga and co-workers (1990) studied harmonic wave propagation in a fluid-loaded composite circular cylinder, where the cylinder was composed of isotropic elastic layers. In their computation, they developed an analytical method to determine the impedance of laminated elastic cylinders. Braga et al. (1990) considered the wave propagation in fluid-loaded laminated cylindrical shells. Hamidzadeh and Sawaya (1995) studied the free vibration of multilayer visco-elastic cylinders. To introduce visco-elastic damping in the composite cylinder, their analysis allowed complex shear moduli for each layer. The problem was expressed in the form of matrix equations relating the displacements of a point in the medium to the boundary stresses. Their analytical approach for a single frequency independent visco-elastic layer was developed to obtain the displacements and stresses in terms of boundary stresses for any given circumferential and axial wave number. This formulation was used to develop a relation between displacements and stresses at the boundaries of each layer and consequently to obtain a propagator matrix relating displacement and stresses of the inner boundaries to the outer ones. Invoking the zero boundary stresses, the analysis led to a frequency equation for the free vibration of the multilayer visco-elastic cylinder. The presented analytical procedure was also
10
1. Introduction
applicable to multilayer cylinders of arbitrary laminated configuration and material damping for each layer. Zhou and Yang (1995 and 1996) developed a distributed transfer function method for vibration analysis of thick laminated composite cylinders. Qatu (1999) employed Dong’s solution (1968) and provided analysis for laminated composite shells. He considered the free vibration analysis and evaluated frequencies for orthotropic and isotropic cylindrical shells. Qing et al. (2006) presented a semi-analytical solution along with a finite element method for free vibration of thick double layer cylinder.
1.8 Vibrations of Constrained Layered Cylindrical Structures Damping is a significant dynamic parameter for vibration, sound control, dynamic stability, positioning accuracy, fatigue endurance, and impact resistance. Many current applications of the circular structure, (aerospace industry, power generation or transmission, heat exchangers, tubes, nuclear reactor components, smoke stacks, and undersea technology) require high dynamic performance. Therefore, candidate sources of passive damping should be compatible with structural configuration. To control large displacement and stresses, damping treatment using visco-elastic material with complex shear modulus is employed. In the constrained layer damping treatment, a visco-elastic material is sandwiched between two elastic layers. The visco-elastic layer dissipates much of the energy within the visco-elastic cylinder in the form of heat during shear de-formation when subjected to harmonic vibration. In this configuration, the core layer allows shear displacement and heat dissipation, while the elastic layers give the cylindrical structure rigidity and stability. The appropriate use of the visco-elastic damping material is to sandwich it between two elastic materials. The elastic material will give the structure stiffness, while the damping material will dissipate much of the energy. With the continuing evolution of the composite structures and simultaneous requirements for higher performance and lower operational costs, more and more composites are targeted toward structural applications involving combined dynamic, mechanical, thermal, and hydraulic loading. Composite structures are principally preferred in such applications because of their advanced elastic properties for individual design requirements. They also have the potential for incorporating significant passive damping into the candidate structure.
1. Introduction
11
The potential damping sources satisfying the previous requirements are the constrained layer damping approach. Constrained damping in isotropic metallic structures has been applied and investigated. The inherent damping capacity of composite structures also seems promising. Although the damping of composite structures is not very high, it is significantly higher than most of the common metallic structures. Moreover, visco-elastic damping layers are the material of preference in many cases, as they readily provide high specific stiffness and strength. Constrained-layer cylindrical structures with visco-elastic damping layers will offer the advantages of high damping and good mechanical properties. In addition, the visco-elastic damping concept is highly compatible with the laminated configuration of composite structures. A visco-elastic layer may produce higher and also tailorable damping. However, other critical mechanical properties, such as stiffness and strength, are expected to be reduced. It is desirable to develop integrated composite mechanics that will efficiently predict the damping of composite structures with a viscoelastic damping layer, and other critical mechanical properties and dynamic characteristics. An analytical work including the damping analysis of threelayered cylinders with constrained inter-laminar layer of visco-elastic material has been reported. The effects of different geometries and damping layer thicknesses are also investigated. Research on the constrained layer damping treatment for beams was conducted by Agbasiere and Grootenhuis (1968). In this study, they transformed the equations of motion for multilayered beams with visco-elastic cores into non-dimensional simultaneous equations and used finite difference methods for their solution. It was shown that having a damping layer in shear strain in between two rigid layers was effective. Grootenhuis (1970) demonstrated the variation of loss-factor, frequency parameter and tip amplitude ratio for a three-layered symmetric cantilever beam. Nakra and Grootenhuis (1972) presented the variation of loss-factor versus shear parameter for a different number of layers in beams and concluded that dual visco-elastic cores are superior to either no core or a single core. Nakra and Grootenhuis (1974) presented the effect of extensional properties on the loss-factor of a beam that is unsymmetrical and multi-layered, where the shear strain across the visco-elastic layer is not constant. Nakra (1978) provided a summary of damping treatments for structures including viscoelastic material use, multi-layered configuration and constrained layered configuration. Vinson and Sierakowski (2004) investigated the damping behavior of structures made of composite materials. Initial research on the vibration of sandwich cylindrical shells was done by White (1960), Yu (1960), and Eienieck and Fraudenthal (1996), who included layers with visco-elastic properties. Jones and Salerno (1965) investigated the damping effect in the forced axisymmetric vibration of a
12
1. Introduction
cylindrical sandwich shell. The sandwich cylinders considered by Jones and Salerno (1977) were infinitely long or simply supported. Pan (1968) studied the axisymmetrical vibrations of a circular sandwich shell using a visco-elastic layer with a complex shear modulus. He derived and presented the partial differential equations for the motion. The solution of these differential equations yields the frequency equation and the corresponding modal loss-factor after satisfying the boundary conditions. DiTaranto et al. (1973) provided experimental and theoretical data for a driving point impedance in damped composite rings. Lu et al. (1973) introduced damping material in the core layer and analyzed the mechanical impedance of damped three-layered sandwich rings by using the approximate method. They considered the effects of the operational temperature and frequency ranges on the visco-elastic material. The experimental data compared reasonably well with their theoretical predictions for lower modes. Lu and Everstine (1980) included a discontinuous layer in their presentation for the mechanical impedance of layered rings and beams. Hamidzadeh and Chandler (1991) analytically simulated one of the cases presented by Lu et al (1973) using forced vibration analysis. Their results compared satisfactorily well with those of Lu et al. (1973). El-Rahed and Wanger (1986) considered the damping response of shells by using a constrained visco-elastic layer. Hamidzadeh et al. (1991) studied the damping of lobar vibrations in cylindrical structures. They developed an analytical solution for the constrained three-layer thick cylinders that has a visco-elastic middle layer. They utilized the stresses and displacements at all interfaces, and complied with the compatibility requirements at each interface. Hamidzadeh and Sawaya (1993) studied the effect of core material loss-factor, and the effect of the outer layer thickness on the natural frequencies and their corresponding modal loss-factor for a three-layer cylinder. Hamidzadeh and Jiang (1995) presented an analytical solution for the free vibration of thick elastic multi-layer cylinders. The presented method determined the natural frequencies and modal loss-factors for different circumferential wave numbers and a number of different geometric configurations. Hamidzadeh (2008) considered the effect of the visco-elastic core thickness on modal loss-factors of a thick three-layer cylinder.
1.9 Vibrations of Cylindrical Panels An important step in the study of cylindrical panels is the evaluation of the modal displacement functions and their corresponding resonant frequencies. This modal information plays a key role in the design of cylindrical panels
1. Introduction
13
subjected to dynamic excitations. Most studies on the dynamic response of cylindrical panels have been restricted to the application of the theory of cylindrical shells. In the present theory, the equation of motion for a cylindrical shell is formed by taking into account the effect of transverse shear deformations and rotary inertia on the natural frequencies of the shell. Hamidzadeh (1997) extended the research to develop the two-dimensional visco-elastic theory involving the cylindrical panels. The solution was developed to get the stresses and displacements for several panel angles, fourflexural mode, and a wide range of thickness modes. The resonant frequency results of cylindrical panels compared satisfactorily with those of complete cylinders. Hamidzadeh and Medhora (1995) considered the free transverse vibration of a composite thick shell with discontinues constraining layer.
2 Governing Equations This chapter develops the governing equations of motion for a homogeneous isotropic elastic solid, using the linear three-dimensional theory of elasticity in cylindrical coordinates. At first, classical relationships between stress, strain, and displacement are reviewed and implemented into the dynamic equilibrium equations. The mathematical representations of the linear theory of elasticity derived in this chapter will set the stage for the development of the required governing equations for the possible modes of vibrations in cylindrical structures with any thickness.
y
σrr σ zz
σ θθ
δz
δr
σ zz σ rr
r
σ θθ
δθ θ
x
z FIGURE 2.1. Direct Stresses in Cylindrical Coordinates.
A detailed mathematical review of elasto-dynamic problems can be found in most classical text books on advanced mechanics of materials and the theory of elasticity. In particular, references such as (Ford and Alexander 1963) and (Mal and Singh 1991) can provide the best extensive reviews H.R. Hamidzadeh, R.N. Jazar, Vibrations of Thick Cylindrical Structures DOI 10.1007/978-0-387-75591-5_2, © Springer Science+Business Media, LLC 2010
16
2. Governing Equations
σθθ +
∂σθθ δθ ∂θ τθr +
∂τθr δθ ∂θ τrθ +
δθ
τ zr +
σ rr
∂τzr δz ∂z
∂τrθ δr ∂r σrr +
∂σrr δr ∂r
τrθ τθr σθθ FIGURE 2.2. Stresses in the r and θ Directions.
of stress, strain, and displacement in cylindrical coordinates. The following sections provide a succinct review of essential topics needed for the establishment of the governing elasto-dynamic equations.
2.1 State of Stresses at a Point A three dimensional state of stress in an infinitesimal cylindrical element is shown in the following three figures. Figure 2.1 depicts such an element with direct stresses, dimensions, and directions of the cylindrical coordinate. Figure 2.2 represents the direct and shear stresses in the radial and transverse directions (r and θ), and the variation of direct and shear stresses in these two directions. Figure 2.3 shows direct and shear stresses associated with the planes perpendicular to the r and z directions, as well as their variations along these directions. In the above graphical representations the changes in direct and shear stresses are given by considering the first order infinitesimal term used in Taylor series approximation. The series approximation has been truncated after the second term. Further terms within the series representation contain terms of an infinitesimal length squared. Assuming that the second
2. Governing Equations
z
σ zz + σrr
17
∂σ zz δz ∂z ∂τ zθ δθ ∂θ ∂τ τ zr + zr δr ∂r Fr
τ zθ + δz
σ θθ
τθr δr τθz
σrr +
∂σrr δr ∂r
δθ
r FIGURE 2.3. Stresses in the plane perpendicular to r and z direction.
order terms are very small, they can be neglected. Therefore, the change in stress across the element is considered very small.
2.2 Equilibrium Equations in Terms of Stress Utilizing Newton’s second law and the graphical representation of the state of stress, the equilibrium equations for an infinitesimal element in a cylindrical coordinates will be developed. By examining the state of stress on the element shown in section 2.1, the following equilibrium equation in the r direction is given. µ ¶ µ ¶ ∂σrr ∂τ rθ δθ σ rr + δr (r + δr) δθδz + τ rθ + δθ δrδz cos ∂r ∂θ 2 µ ¶µ ¶ ∂τ rz δr + τ rz + δz r+ δrδθ + Fr δrδθδz ∂z 2 µ ¶ δθ δr = σrr rδθδz + δτ rθ δrδz cos + τ rz r + δrδθ + 2 2 µ ¶ ∂σθθ δθ δθ + σθθ + δθ δrδz sin + σ θθ δrδz sin (2.1) ∂θ 2 2
18
2. Governing Equations
Canceling appropriate terms from both sides of the equation and after simplifying, it yields: ∂σ rr 1 ∂τ rθ ∂τ rz σ rr − σ θθ + + + + Fr = 0 ∂r r ∂θ ∂z r
(2.2)
Similarly, the equilibrium equation for the θ direction yields: 1 ∂σ θθ ∂τ θz 2 ∂τ rθ + + + τ rθ + Fθ = 0 ∂r r ∂θ ∂z r
(2.3)
and finally, for the z direction one may write: ∂τ rz 1 ∂τ θz ∂σ zz 1 + + + τ rz + Fz = 0 ∂r r ∂θ ∂z r
(2.4)
In the above simplifications, due to very small angle of δθ, the following approximations were used: cos
δθ ≈1 2
sin
δθ δθ ≈ 2 2
(2.5)
In addition to the stresses, body forces acting throughout the element have been considered for each direction. These are denoted by Fr , Fθ , and Fz which are introduced as forces in the r, θ, and z direction per unit of volume. Due to the cancellation of the moments about each of the three perpendicular axes, the relations among the six shear stress components are presented by the following three equations: τ rθ = τ θr
τ θz = τ zθ
τ zr = τ rz
(2.6)
Therefore, the stress at any point in the cylinder may be accurately described by three direct stresses and three shear stresses.
2.3 Stress-Strains Relationships The constitutive relation between stresses and strains for a homogeneous and isotropic material can be expressed by Hooke’s law. By definition, a homogeneous and isotropic material has the same properties in all directions. From this, the following three equations for direct strain in terms of stress are presented: err E eθθ E ezz E
= σ rr − ν (σ θθ + σ zz ) = σ θθ − ν (σzz + σ θθ ) = σ zz − ν (σ rr + σ θθ )
(2.7) (2.8) (2.9)
2. Governing Equations
19
where err , eθθ , and ezz are the direct strain in the r, θ, and z directions respectively; E is the Young’s modulus or the modulus of elasticity; and ν is a proportionality factor called Poisson’s ratio. The other three Hooke’s law relations result from the following proportionality between shear stresses and shear strains: τ rθ τ rz τ θz
= Gerθ = Gerz = Geθz
(2.10) (2.11) (2.12)
where, erθ is the shear strain along θ and perpendicular to r; erz is the shear strain along z and perpendicular to r; eθz is the shear strain along z and perpendicular to θ; and G is the shear modulus or the modulus of rigidity. Through the general definition of shear stress and strain, the relationship between shear modulus, Young’s modulus, and Poisson’s ratio is given as: G=
E 2 (1 + ν)
(2.13)
Lame’s elastic constant, λ, and volumetric strain, ε are introduced by the following equations: νE (1 − 2ν) (1 + ν) ε = err + eθθ + ezz
λ =
(2.14) (2.15)
Then, a different form of Hooke’s law relating direct stresses and direct strains can be achieved by adding the direct strain equations (2.7)-(2.9). Eε = σ rr + σ θθ + σ zz 1 − 2ν
(2.16)
Rearranging equation (2.7), one may write: σ θθ + σ zz =
σ rr − Eerr ν
(2.17)
Now, by substituting equation (2.17) into (2.16) it yields: Eε σ rr Eerr = σ rr + − 1 − 2ν ν ν
(2.18)
and the result can be arranged as: νEε = (1 + ν) σ rr − Eerr 1 − 2ν
(2.19)
20
2. Governing Equations
from which the direct stress in the radial direction is determined to be: σ rr =
νE E ε+ err (1 − 2ν) (1 + ν) 1+ν
(2.20)
Now using the definitions of the shear modulus and Lame’s elastic constant, the direct radial stress is presented as: σ rr = λε + 2Gerr
(2.21)
In a similar procedure, the direct circumferential stress and the direct axial stress are represented in terms of the volumetric strain, Lame’s elastic constant, the shear modulus, and the appropriate direct strains are presented in the following equations. σ θθ σzz
= λε + 2Geθθ = λε + 2Gezz
(2.22) (2.23)
2.4 Strain-Displacement Relationships In Figure 2.4, a small element of an elastic homogenous and isotropic medium is represented in cylindrical coordinates. The element contains the point A, which represents a given point having the coordinates of (r, θ, z) and the point F , an infinitesimal distance away, having the coordinates (r + δr, θ + δθ, z + δz). In this figure, the angle θ may be measured from any arbitrary coordinate direction such as x. A typical small linear deformation of this element is depicted in Figure 2.5 where displacement and the distorted shape of the enlarged element is outlined. As can be seen, the displacement of point A to A0 is defined by the three components of ur , uθ , and uz . Where ur , uθ , and uz are the displacements in the radial direction, transverse direction, and axial direction, respectively. It should be noted that uθ is the actual linear displacement along a circumferential arc. The displacements of the point F to F 0 are (ur + δur ), (uθ + δuθ ), and (uz + δuz ). Considering a horizontal plane, in Figure 2.6, the face ACDB of the element previously shown in Figure 2.5 moves to A0 C 0 D0 B 0 where there is a change in the length of the sides and the angles are sheared. Angle shearing is resulted by the change of the angle δθ to (δθ + 4δθ).
2. Governing Equations
z y H
δz
G
D
E
A
θ
F ( r + δr , θ + δθ, z + δz )
B
C
δθ r δr
x
FIGURE 2.4. An Element in Cylindrical Coordinates.
z F1
H F
G
uθ
x
B
uz
A ur
F'
E
A'
D' D1 C1
C'
D
C FIGURE 2.5. Element Subjected to Small Deformation.
y
21
22
2. Governing Equations
D' B1 B '
D
B
O
δθ + Δδθ
δθ
uθ r
C' C3 C1
A'
A A 2 D
C2
FIGURE 2.6. Horizontal Plane of Strained Element.
Upon examining the radial strain at point A and ignoring the effects of strains in the z direction, the strain in the side AC can be found. If the distance A0 C 0 is transferred to the line AC by drawing arcs, with center O, through A0 and C 0 to intersect the line OAC at points A2 and C2 then the radial strain can be defined as: err =
A2 C2 − AC AC
(2.24)
Considering the geometry of the Figure 2.6, the above equation can be written in the following form: µ ¶ ∂ur δr + δr − δr ∂ur ∂r = (2.25) err = δr ∂r In a similar manner, the direct circumferential strain may be defined as: eθθ =
A0 B 0 − AB AB
(2.26)
where, AB = rδθ and
µ ¶ ∂uθ δθ . A B = (r + ur ) δθ + r∂θ 0
0
(2.27)
In the definition of A0 B 0 , the increase in the angle of the arc of A0 B 0 is given by δuθ / (r + ur ) which can be approximated by (∂uθ / (r∂θ)) δθ.
2. Governing Equations
23
Therefore, by neglecting the second order terms, the circumferential direct strain is given by: ∂uθ ur eθθ = + (2.28) r∂θ r The shear strain, erθ , is represented by the change of the angle BAC. By drawing A0 C1 parallel to AC, A0 B1 parallel to AB, and continuing line OA0 to yield point C3 , the following procedures will yield erθ which is a rate of change of the line A0 C1 . Notice that A0 C1 is parallel to line AC.
where,
erθ = ∠C3 A0 C 0 + ∠B1 A0 B 0
(2.29)
∠C3 A0 C 0 = ∠C1 A0 C 0 − ∠C1 A0 C3
(2.30)
and ∠C1 A0 C3 = ∠AOA0 =
A2 A0 uθ = . r + ur r
(2.31)
The definition of the length C1 C 0 is: C1 C 0 =
∂uθ δr ∂r
(2.32)
then,
C1 C 0 ∂uθ = . AC ∂r Therefore, the shear angle C3 A0 C1 can be defined as: ∠C1 A0 C 0 =
∂uθ δr u ∂uθ uθ θ ∠C3 A0 C 0 = ∂r − = − δr r ∂r r
(2.33)
(2.34)
The shear angle B1 A0 B 0 can be defined as: ∠B1 A0 B 0 =
B1 B 0 rδθ
(2.35)
which is the radial displacement of B due to the angle δθ over the initial length. In partial derivative form this simplifies to: ∠B1 A0 B 0 =
1 ∂ur ∂ur δθ = rδθ ∂θ r∂θ
(2.36)
Therefore, the shear strain erθ , is given as: erθ =
∂uθ uθ ∂ur − + ∂r r r∂θ
(2.37)
24
2. Governing Equations
H'
G' G
1
H
G δz
B' B1
A'
B
rδθ
A
FIGURE 2.7. The (z, θ) Plane of Strained Element.
Considering the z direction, the direct axial strain can be defined similarly to the procedure used in Cartesian coordinates. Recall that the axial strain is defined as the ratio of the change in length to the original length of the element in the z direction. Examining the strain in line AF in the z direction, the direct axial strain is given as: ezz =
A0z Fz0 − Az Fz Az Fz
(2.38)
which simplifies to: δz + ezz =
∂uz δz − δz ∂z δz
(2.39)
and finally to: ezz =
∂uz ∂z
(2.40)
In Figure 2.7 the z plane is shown as viewed from the origin. On the face ABHG the shear strain eθz causes the right angle BAG to be displaced to B 0 A0 G0 . Note that A0 B1 is parallel to AB and A0 G0 is parallel to AG. Therefore, the shear strain, eθz , is given as: eθz = ∠G1 A0 G0 + ∠B1 A0 B 0
(2.41)
1 ∂uθ 1 ∂uz δz + δθ δz ∂z rδθ ∂θ
(2.42)
∂uθ ∂uz + ∂z r∂θ
(2.43)
which is equivalent to: eθz = and simplifies to: eθz =
2. Governing Equations
25
Finally by examining the (r, z) plane, the shear strain, erz , is defined as: erz = ∠G1 A0 G0 + ∠C1 A0 C 0
(2.44)
which yields: erz
∂ur ∂uz δz δr ∂z = + ∂r δz δr
(2.45)
and simplifies to: erz =
∂ur ∂uz + ∂z ∂r
(2.46)
There are now six strain components given in terms of the cylinder displacements. This completes the development of the required strain-displacement relationships.
2.5 Stress-Displacement Relationships In this section, the stress-displacement relationships are developed by building upon Hooke’s law and strain displacement relationships. Beginning with the direct radial stress in terms of strain and substituting the equations for direct strains, the radial stress in terms of displacement can be presented as: µ ¶ ∂ur ∂uθ ur ∂uz ∂ur σrr = λ + + + + 2G (2.47) ∂r r∂θ r ∂z ∂r The substituted direct strains are in terms of displacements and the volumetric strain. Similarly, the direct circumferential stress and direct axial stress, in terms of displacement, may be given as: σ θθ σ zz
µ
¶ µ ¶ ∂ur ∂uθ ∂uθ ur ∂uz ur = λ + + + + 2G + ∂r r∂θ r ∂z r∂θ r µ ¶ ∂ur ∂uθ ur ∂uz ∂uz = λ + + + + 2G ∂r r∂θ r ∂z ∂z
(2.48) (2.49)
Similarly, the three shear stresses in terms of shear strains are given by equations (2.10)-(2.12) and the shear strains, in terms of displacement components, are provided by equations (2.37), (2.43), and (2.46). Therefore,
26
2. Governing Equations
these shear stresses are: τ rθ τ θz τ rz
µ
∂uθ uθ ∂ur = G − + ∂r r r∂θ µ ¶ ∂uθ ∂uz = G + ∂z r∂θ µ ¶ ∂ur ∂uz = G + ∂z ∂r
¶
(2.50) (2.51) (2.52)
2.6 Equations of Motion In this section, the governing equations of motion in terms of a displacement vector are generated. The displacement vector is given as: u = urˆır + uθˆıθ + uzˆız
(2.53)
where ˆır , ˆıθ , and ˆız denote unit vectors directed along the (r, θ, and z) axes, respectively. Substituting Hooke’s law equations into the dynamic equilibrium equations and introducing strain-displacement relationships yield the governing equations of motion: ∂ε ∂ 2 ur (2.54) = ρ 2 ∂r ∂t ∂ε ∂ 2 uθ (2.55) μ∇2 uθ + (λ + μ) = ρ 2 r∂θ ∂t ∂ε ∂ 2 uz (2.56) μ∇2 uz + (λ + μ) = ρ 2 ∂z ∂t where μ is the same as shear modulus G, ε is the volumetric strain, and the ∇2 is the three dimensional Laplacian operator in cylindrical coordinates defined by: ∂2 ∂ ∂2 ∂2 ∇2 = 2 + (2.57) + 2 2+ 2 ∂r r∂r r ∂θ ∂z Multiplying equation (2.54) by ˆır , equation (2.55) by ˆıθ , equation (2.56) by ˆız , and adding these three equations, the vector form of the governing equation of motion is given by: μ∇2 ur + (λ + μ)
∂2u ∂t2
(2.58)
where ∇ is the cylindrical gradient: µ ¶ ∂ ∂ ∂ 1 ∇= + ˆır + ˆıθ + ˆız ∂r r r∂θ ∂z
(2.59)
μ∇2 u + (λ + μ) ∇ (∇ · u) = ρ
2. Governing Equations
2.7 Key Symbols A, B, · · · A0 , B 0 e err , eθθ , ezz erθ erz eθz E Fr , Fθ , Fz G, μ ˆır , ˆıθ , ˆız r ur , uθ , uz u z
point label strain direct strain in r, θ, z directions shear strain along θ and perpendicular to r shear strain along z and perpendicular to r shear strain along z and perpendicular to θ Young’s modulus forces in the r, θ, z directions per unit of volume shear modulus, modulus of rigidity unit vectors along the axes r, θ, z radial direction displacements in r, θ, z directions displacement vector axial direction
δ ε θ λ μ, G ν σ σ rr , σ θθ , σzz τ τ rθ τ rz τ θz
variation volumetric strain transverse direction Lame’s elastic constant shear modulus, modulus of rigidity Poisson’s ratio normal stress direct stress in the r, θ, z directions shear stress shear stress along θ and perpendicular to r shear stress along z and perpendicular to r shear stress along z and perpendicular to θ
∠ ∇2 ∇
angle Laplacian operator gradient
27
3 Vibration of Single-Layer Cylinder The following describes the procedures used to develop a mathematical model for the vibration analysis of a single-layer cylinder. The medium for the cylinder is assumed to be a homogeneous and isotropic elastic with infinite length. A finite length cylinder can only be directly analyzed for the cylinders with simply supported end conditions.
b θ
a
z
r FIGURE 3.1. Reference Coordinates and Dimensions.
The presented mathematical procedure in this chapter is based on the work discussed by Armenakas et al. (1969), for the free vibration case. Their technique was refined by Hamidzadeh et al. (1981) to develop an analytical approach for studying the free, as well as, forced vibrations for these structures. The presented mathematical procedure implements the linear elasto-dynamic theory and formulates the displacements and stresses to satisfy the required boundary conditions for both cases of free and forced vibrations. Figure 3.1 describes the cylindrical coordinates and the required geometrical parameters.
H.R. Hamidzadeh, R.N. Jazar, Vibrations of Thick Cylindrical Structures DOI 10.1007/978-0-387-75591-5_3, © Springer Science+Business Media, LLC 2010
30
3. Vibration of Single-Layer Cylinder
3.1 Governing Equation The governing equation of motion for a cylinder in invariant form was developed and presented in equation (2.58) in the previous chapter, and for continuity is repeated here. μ∇2 u + (λ + μ) ∇ (∇ · u) = ρ
∂2u ∂t2
(3.1)
where u is the displacements vector, ρ is the density of the medium, λ and μ are Lamé elastic constants, and ∇2 is the three dimensional Laplacian operator. The boundary conditions for these structures can be written in terms of boundary stresses in both inner and outer surfaces of the cylinder. These stress components are σrr , τ rθ , τ rz . They are applied at both the inner surface (r = a) and the outer surface (r = b) boundaries. Each of these boundary stresses is considered to be the source of excitations for forced vibration of the cylindrical structures.
3.2 Proposed Solution The governing partial differential equations of motion can be solved using Lamé potential functions φ and H, where the displacements vector u in terms of these potential functions is presented by the following equation. u = ∇φ + ∇ × H
(3.2)
H = Hrˆır + Hθˆıθ + Hzˆız .
(3.3)
where
In the following section, it will be shown that the above displacements vector given in terms of the Lamé potential functions φ and H, will satisfy the governing equations of motion. Moreover, it will be proven that these potential functions can be presented using the following wave equations. v12 ∇2 φ = v12 ∇2 H =
∂2φ ∂t2 ∂2H ∂t2
(3.4) (3.5)
3. Vibration of Single-Layer Cylinder
31
where, v1 and v2 are the velocities of propagation of the dilation and distortional wave, respectively. They can be presented as: s λ + 2μ v1 = (3.6) ρ r μ v2 = (3.7) ρ
3.3 Derivation of Wave Equation By using Lamé potential functions, the governing equation of motion is reduced to two wave equations. These equations are developed in the following series of equations. First, substituting equation (3.2) into the governing equation of motion (3.1) yields: μ∇2 (∇φ + ∇ × H) + (λ + μ) ∇∇ · (∇φ + ∇ × H) ∂2 = ρ 2 (∇φ + ∇ × H) ∂t
(3.8)
Expanding the above equation yields: μ λ μ μ 2 ∇ ∇φ + ∇2 (∇ × H) + ∇∇ · ∇φ + ∇∇ · ∇φ ρ ρ ρ ρ λ μ + ∇∇ · (∇ × H) + ∇∇ · (∇ × H) ρ ρ 2 ∂2 ∂ = 2 ∇φ + 2 (∇ × H) ∂t ∂t
(3.9)
Now, separating the terms containing the potential functions develops two independent equations in terms of φ and H. Applying the property of ∇2 f = ∇ · ∇f , the first equation can be simplified in the form: λ μ ∂2 μ ∇∇2 φ + ∇∇2 φ + ∇∇2 φ = 2 ∇φ ρ ρ ρ ∂t
(3.10)
After rearranging and simplifying, the first wave equation will be written as: ∂2φ λ + 2μ 2 (3.11) ∇ φ= 2 ρ ∂t Substituting equation (3.6) into the above equation yields: v12 ∇2 φ =
∂2φ ∂t2
(3.12)
32
3. Vibration of Single-Layer Cylinder
The remainder of the equation (3.9) in terms of H can be simplified into the second wave equation. λ+μ ∂2 μ 2 ∇ (∇ × H) + ∇∇ · (∇ × H) = 2 (∇ × H) ρ ρ ∂t
(3.13)
Applying the vector property of ∇ · (∇ × V) = 0 (for any arbitrary vector of V), the equation (3.13) reduces to: ∂2 μ 2 ∇ (∇ × H) = 2 (∇ × H) ρ ∂t
(3.14)
After simplifying and substituting equation (3.7) it yields: v22 ∇2 H =
∂2 H ∂t2
(3.15)
This is the second wave equation. Considering Lamé potential functions, equations (3.12) and (3.15) are a modified form of the governing equations of motion.
3.4 Solutions to the Wave Equations Throughout this section, the assumed solutions to the two modified governing equations (3.12) and (3.15) will be written in the following forms:
φ Hr Hθ Hz
= = = =
f (r) cos (nθ) cos (ωt + ζz) gr (r) sin (nθ) sin (ωt + ζz) gθ (r) cos (nθ) sin (ωt + ζz) g3 (r) sin (nθ) cos (ωt + ζz)
(3.16) (3.17) (3.18) (3.19)
In the following sections it will be proven that the functions f (r), gr (r), gθ (r), and g3 (r) are solutions to the following Bessel differential equations Bn,αx [f ] Bn,βr [g3 ] Bn+1,βr [gr − gθ ] Bn−1,βr [gr + gθ ]
= = = =
0 0 0 0
(3.20) (3.21) (3.22) (3.23)
3. Vibration of Single-Layer Cylinder
where the Bessel Differential Operator Bn,x is defined as: ¶ µ 2 d2 d n −1 − Bn,x = 2 + dx xdx x2
33
(3.24)
and n is an integer number.
3.4.1 Derivation of Bessel Equation Bn,αx [f] = 0 The first wave equation was given in the following form: v12 ∇2 φ −
∂2 φ=0 ∂t2
(3.25)
where: ∂2 ∂ ∂2 ∂2 + + + 2 ∂r2 r∂r r2 ∂θ ∂z 2 φ = f (r) cos (nθ) cos (ωt + ζz)
∇2
=
(3.26) (3.27)
Substituting equations (3.26) and (3.27) into the first wave equation (3.25) and dividing by the harmonic term cos (nθ) cos (ωt + ζz) yields: µ 2 ¶ ¶ µ 2 d f (r) df (r) n 2 2 v1 + + ζ f (r) + ω 2 f (r) = 0 (3.28) − dr2 rdr r2 Rearranging the terms yields: d2 f (r) df (r) n2 f (r) + + − dr2 rdr r2
µ
¶ ω2 2 − ζ f (r) = 0 v12
(3.29)
By defining α as: ω2 − ζ2 v12 and substituting into equation (3.29) yields: α2 =
d2 f (r) df (r) n2 f (r) + + α2 f (r) = 0 − dr2 rdr r2 Equation (3.31) can be arranged into the following two steps: d2 f (r) df (r) n2 f (r) + 2 − 2 2 + f (r) = 0 2 2 α dr α rdr α r ! Ã d2 f (r) df (r) n2 2 + αrd (αr) − 2 − 1 f (r) = 0 d (αr) (αr)
(3.30)
(3.31)
(3.32) (3.33)
Here, equation (3.33) is the first Bessel differential equation, presented in equation (3.20).
34
3. Vibration of Single-Layer Cylinder
3.4.2 Derivation of the Bessel Equation Bn,βr [gz ] = 0 The second wave equation was presented by equation (3.15) and it can be separated into three scalar equations in terms of Hr , Hθ , and Hz . The scalar equation in terms of Hz can be presented as: v22 ∇2 Hz −
∂2 Hz = 0 ∂t2
(3.34)
where Hz is given by: Hz = g3 (r) sin (nθ) cos (ωt + ζz)
(3.35)
Substituting Hz from (3.35) into (3.34) and dividing the results by the harmonic function sin (nθ) cos (ωt + ζz) yields: d2 g3 (r) dg3 (r) n2 g3 (r) + + − dr2 rdr r2
µ
¶ ω2 2 − ζ g3 (r) = 0 v22
(3.36)
By defining β as: β2 =
ω2 − ζ2 v22
(3.37)
Substituting equation (3.37) into (3.36) after rearrangement and simplification yields: ! Ã dg3 (r) d2 g3 (r) n2 + − 1 g3 (r) = 0 (3.38) − βrd (βr) d (βr)2 (βr)2 Equation (3.38) is the second Bessel differential equation, which was presented by equation (3.21).
3.4.3 Derivation of the Bessel Equations Bn+1,βr [gr − gθ ] = 0 and Bn−1,βr [gr + gθ ] = 0 Let us begin with the second wave equation, v22 ∇2 H −
∂2 H=0 ∂t2
(3.39)
and reintroducing functions of Hr and Hθ as follows. Hr Hθ
= gr (r) sin (nθ) sin (ωt + ζz) = gθ (r) cos (nθ) sin (ωt + ζz)
(3.40) (3.41)
3. Vibration of Single-Layer Cylinder
35
Then, substituting Hr into the wave equation (3.39) and dividing through by the harmonic term sin (nθ) sin (ωt + ζz), after rearranging the result yields: µ 2 ¶ d2 gr (r) dgr (r) n2 gr (r) ω 2 + + − ζ (3.42) gr (r) = 0 − dr2 rdr r2 v22 Introducing β from equation (3.37) to equation (3.42) yields a new Bessel differential equation. ! Ã dgr (r) n2 d2 gr (r) + − 1 gr (r) = 0 (3.43) − βrd (βr) d (βr)2 (βr)2 which can be presented in the following form: Bn,βr [gr (r)] = 0
(3.44)
where, the constant n can be any integer value. Therefore, the following two equations are also acceptable: Bn+1,βr [gr (r)] = 0 Bn−1,βr [gr (r)] = 0
(3.45) (3.46)
Next, by substituting Hθ from equation (3.41) into the wave equation (3.39) and dividing through by the harmonic term yields: µ 2 ¶ d d n2 2 2 v2 + (3.47) − 2 − ζ gθ (r) + ω 2 gθ (r) = 0 dr2 rdr r Again, by introducing β from equation (3.37) to equation (3.47) yields an additional Bessel differential equation in the following form: ! Ã dgθ (r) n2 d2 gθ (r) (3.48) 2 + βrd (βr) − 2 − 1 gθ (r) = 0 d (βr) (βr) This equation can be presented in the following Bessel Differential Operator form (3.49) Bn,βr [gθ (r)] = 0 Again, n can be any integer value such as n+1 or n−1. Therefore, equation (3.49) can be written by the next two equations: Bn+1,βr [gθ (r)] = 0 Bn−1,βr [gθ (r)] = 0
(3.50) (3.51)
36
3. Vibration of Single-Layer Cylinder
Subtracting equation (3.50) from equation (3.45) results in the third Bessel differential equation, Bn+1,βr [gr ] − Bn+1,βr [gθ ] = Bn+1,βr [gr − gθ ] = 0.
(3.52)
Adding equation (3.51) and (3.46) results in the fourth Bessel differential equation, Bn−1,βr [gr ] + Bn−1,βr [gθ ] = Bn−1,βr [gr + gθ ] = 0.
(3.53)
3.5 Solutions to the Bessel differential equations The general solutions to the Bessel differential equations, (3.20)-(3.23), are discussed by many such as Watson (1962), Korenev (2002), Abramowitz and Stegun (1965), and Bell (1968). The solutions to the above mentioned differential equations are presented in the combination of the first and second kinds of Bessel functions by the following equations, f (r) g3 (r) gr (r) − gθ (r) gr (r) + gθ (r)
= = = =
A1 Jn (αr) + B1 Yn (αr) A2 Jn (βr) + B2 Yn (βr) A3 Jn+1 (βr) + B3 Yn+1 (βr) A4 Jn−1 (βr) + B4 Yn−1 (βr) .
(3.54) (3.55) (3.56) (3.57)
However, to simplify the equations for stresses and displacements, the following definitions are given: 2g1 (r) = gr (r) − gθ (r) 2g2 (r) = gr (r) + gθ (r)
(3.58) (3.59)
It can be shown that any one of the three potential functions (i.e. g1 , g2 , or g3 ) can be set equal to zero without the loss of the generality of the solutions, (Markus, 1988). Therefore, in this text the value of g2 has been set to zero.
3.6 Displacements Solving the resulted four wave equations, the solution to the Lamé potential functions φ, Hr , Hθ , and Hz are achieved. Furthermore, by substituting these potential functions into equation (3.2), the components of the
3. Vibration of Single-Layer Cylinder
37
displacement in the r, θ, z directions at any point within the medium of the cylindrical structure are determined by the following equations: ³ ´ n f 0 + g3 + ζg1 cos (nθ) cos (ωt + ζz) (3.60) ur = ³ n r ´ − f + ζg1 − g30 sin (nθ) cos (ωt + ζz) (3.61) uθ = µ r ¶ n+1 g1 cos (nθ) sin (ωt + ζz) uz = −ζf − g10 − (3.62) r where, primes indicate differentiation with respect to r.
3.7 Stresses The stress equations are developed by substituting the displacement equations into the stress-displacement relations. The resultant direct and shear stresses of interest are: ³ ´´ ³ ¡ ¢ n³ 0 g3 ´ −λ α2 + ζ 2 f + 2μ f 00 + (3.63) h1 σ rr = g3 − + ζg10 r r µ µ ¶ 2n f g0 n2 − τ rθ = f0 − − 2 g3 − g300 + 3 r r r r ¶ ζ nζ (3.64) − g1 + ζg10 − g1 μh2 r r µ µ ¶ ¶ n+1 n+1 0 nζ 2 00 τ rz = −2ζf 0 + − ζ − − g − g g g 1 3 μh3 (3.65) 1 1 r2 r r where, primes indicate derivation with respect to r, and h1 , h2 , h3 are defined by the following equations. h1 h2 h3
= cos (nθ) cos (ωt + ζz) = sin (nθ) cos (ωt + ζz) = cos (nθ) sin (ωt + ζz)
(3.66) (3.67) (3.68)
Although all six stress components can be derived in this form, nevertheless, only those stresses presented in equations (3.63)-(3.65) are needed for the analysis. In fact, for boundary stresses, only derivations of those stresses which occur on the inner and outer surfaces are presented in the next section.
38
3. Vibration of Single-Layer Cylinder
3.7.1 Derivation of the Radial Stress σ rr This section outlines the development of the radial stress component which was introduced in equation (2.47) as: µ ¶ ∂ur ∂uθ ur ∂uz ∂ur σ rr = λ + + + + 2μ (3.69) ∂r r∂θ r ∂z ∂r Substituting the displacement equations (3.60)-(3.62), the radial stress can be presented as: µ ¶ f0 n2 σ rr 2 00 = λ f + − 2f −ζ f h1 r r ³ ´ n³ 0 g3 ´ 00 +2μ f + (3.70) g3 − + ζg10 r r
By adding and subtracting ω 2 f /v12 to the first bracket of equation (3.63), the above equation can be arranged into the following equation µ µ 2 ¶ ¶ σ rr f0 ω ω2 n2 2 = λ f 00 + − ζ f f − − 2f + h1 r r v12 v12 ³ ´ ³ ´ n 0 g3 +2μ f 00 + (3.71) g − + ζg10 r 3 r or µ ¶ σ rr f0 ω2 n2 00 2 = λ f + − 2f +α f − 2f h1 r r v1 ³ ´ n³ 0 g3 ´ 00 +2μ f + (3.72) g3 − + ζg10 . r r
Considering equation (3.24) and (3.20) the above equation will reduce to: µ ¶ ³ ´ ω2 n³ 0 g3 ´ σ rr = λ Bn,αr [f ] − 2 f + 2μ f 00 + (3.73) g3 − + ζg10 h1 v1 r r The Bessel differential equation in the first term is equal to zero. Therefore, equation (3.73) reduces to: ³ ´ σ rr ω2 n³ 0 g3 ´ = −λ 2 f + 2μ f 00 + (3.74) g3 − + ζg10 h1 v1 r r
Substituting in terms of α and ζ for (ω/v1 )2 from (3.30) into the first term of equation (3.74), the final form of the direct radial stress is given as: ¡ ¢ σ rr = −λ α2 + ζ 2 f cos (nθ) cos (ωt + ζz) ³ ´ n³ 0 g3 ´ +2μ f 00 + g3 − + ζg10 cos (nθ) cos (ωt + ζz) (3.75) r r
where, prime indicates differentiation with respect to r.
3. Vibration of Single-Layer Cylinder
39
3.7.2 Derivation of the Shear Stress τ rθ The tangential shear stress in terms of the displacements was presented by equation (2.50) as: µ ¶ ∂ur ∂uθ uθ τ rθ = μ + − (3.76) r∂θ ∂r r Substitution of the displacement equations (3.60)-(3.62) into the above equation yields: ´ ³n ´ n³ 0 n n τ rθ = −μ f + g3 + ζg1 + μ 2 f − f 0 + ζg10 + g300 h2 r r r r ¶ µ ζ g30 n (3.77) −μ − 2 f + g1 − r r r Common terms are grouped and the final form of the tangential shear stress τ rθ is presented by: µ ¶ µ 2 ¶ τ rθ 2n f g0 n = −μ f0 − + μ − 2 g3 − g300 + 3 h2 r r r r ¶ µ ζ nζ (3.78) +μ − g1 + ζg10 − g1 r r where, prime indicates differentiation with respect to r.
3.7.3 Derivation of the Shear Stress τ rz The tangential shear stress τ rz was presented by the equation (2.52) as: µ ¶ ∂ur ∂uz τ rz = μ + (3.79) ∂z ∂r Substituting the displacement equations (3.60)-(3.62), the expanded form of equation (3.79) yields: ³ ´ τ rz n = −μζ f 0 + g3 + g1 h3 r µ ¶ n+1 n+1 0 0 00 +μ −ζf − g1 + g1 − g (3.80) r2 r2 1 Grouping common terms, simplifies the tangential shear stress τ rz to: µ µ ¶ ¶ τ rz n+1 2 = μ −2ζf 0 + − ζ g1 h3 r2 µ ¶ n+1 nζ +μ − 2 g10 − g100 − (3.81) g3 r r
40
3. Vibration of Single-Layer Cylinder
3.8 The Boundary Stresses By entering the solutions to the Bessel differential equations into the three stress components, equations (3.63)-(3.65) can be written in terms of the Bessel functions and their constant coefficients. Assuming arbitrary boundary condition values at the inner and outer surfaces of the cylinder, a linear non-homogeneous system of equations is developed. The matrix form of this system can be expressed by the following relation: TX = S where, X is:
⎧ ⎫ A1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ B1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ A2 X= ⎪ ⎪ B2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ A3 ⎪ ⎪ ⎩ ⎭ B3
(3.82)
(3.83)
and S is the vector of boundary stresses presented by the following equation: ⎫ ⎧ σ rr (r = a) /h1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ τ rθ (r = a) /h2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ τ rz (r = a) /h3 (3.84) S= σ rr (r = b) /h1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ τ rθ (r = b) /h2 ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ τ rz (r = b) /h3
The elements in the first three rows of the T matrix are developed by substituting r = a for the inner boundary radius in equations (3.63)-(3.65). These elements are: ¡ ¢ T1,1 = −λ α2 + ζ 2 Zn (αr) + 2μα2 Zn00 (αr) (3.85) ¡ 2 ¢ 2 T1,2 = −λ α + ζ Wn (αr) + 2μα2 Wn00 (αr) (3.86) 0 T1,3 = 2μζβZn+1 (βr) (3.87) T1,4 T1,5 T1,6 T2,1 T2,2 T2,3
0 = 2μζβWn+1 (βr) n 0 n = 2μ βZn (βr) − 2μ 2 Zn (βr) r r n n = 2μ βWn0 (βr) − 2μ 2 Wn (βr) r r
n n = −2μ αZn0 (αr) + 2μ 2 Zn (αr) r r n n 0 = −2μ αWn (αr) + 2μ 2 Wn (αr) r r n+1 0 = −μζ (βr) Zn+1 (βr) + μζβWn+1 r
(3.88) (3.89) (3.90) (3.91) (3.92) (3.93)
3. Vibration of Single-Layer Cylinder
n+1 0 Wn+1 (βr) + μζβWn+1 (βr) r n2 μ = −μ 2 Zn (βr) − μβ 2 Zn00 (βr) + βZn0 (βr) r r n2 μ = −μ 2 Wn (βr) − μβ 2 Wn00 (βr) + βWn0 (βr) r r
T2,4
= −μζ
T2,5 T2,6 T3,1 T3,2 T3,3
= −2μζαZn0 (αr) = −2μζαWn0 (αr) ¶ µ n+1 n+1 0 2 = μ − ζ Zn+1 (βr) − μ βZn+1 (βr) r2 r 00 −μβ 2 Zn+1 (βr)
T3,4
= μ
µ
41
(3.94) (3.95) (3.96) (3.97) (3.98)
(3.99)
¶ n+1 n+1 2 0 − ζ Wn+1 (βr) − μ (βr) βWn+1 r2 r
00 −μβ 2 Wn+1 (βr) (3.100) n T3,5 = −μζ Zn (βr) (3.101) r n T3,6 = −μζ Wn (βr) (3.102) r where primes denote differentiation with respect to the argument of the Bessel function. Also, Zn and Wn denote the Bessel functions Jn and Yn or the modified Bessel functions In and Kn . The type of Bessel function Zn and Wn represent, depends on whether the argument is real or imaginary. The difference in sign between the recursion formula of the Bessel functions and the modified Bessel functions is handled within the Fortran code. Therefore, no modification of the given T matrix is needed. The proper selection of Bessel functions is given in table 3.1.
Table 3.1 - Bessel functions Interval ω > v1 ζ v1 ζ > ω > v2 ζ ω < v2 ζ
for Different Intervals of the Frequency. Functions Used J (αr) , Y (αr) , J (βr) , Y (βr) I (αr) , K (αr) , J (βr) , Y (βr) I (αr) , K (αr) , I (βr) , K (βr)
Conversion of Bessel function derivatives is conducted according to the following steps. d f dr d f dr d f dr
d (A1 Jn (αr) + B1 Yn (αr)) dr d d = αA1 Jn (αr) + αB1 Yn (αr) αdr αdr d d = αA1 Jn (αr) + αB1 Yn (αr) d (αr) d (αr) =
(3.103) (3.104) (3.105)
42
3. Vibration of Single-Layer Cylinder
The last three rows of the T matrix are formed by substituting r = b for the outer boundary radius. In the case of free vibration, the boundary stresses are zero. Therefore, the determinant of the matrix T must be zero. By solving the resulted equation, the natural frequencies for all modes of free vibration will be determined. In the case of forced vibration, when the known modal boundary stress components are input into equation (3.82), the vector X is calculated. The vector X is then substituted into equations (3.60)-(3.65) to determine all six modal displacements and stresses at any point within the medium.
3.9 The Stress-Displacement Matrix Equation From the stress matrix and the known boundary conditions, the coefficient vector X can be determined, as shown by the following equation: X = T −1 S
(3.106)
Utilizing the known displacement and stress functions from equations (3.60)(3.65) and substituting the Bessel differential equation solutions, a new system of linear equations can be built. The matrix form of this system can be presented by: DX = C (3.107) where the vector X is known from the equation (3.106). The unknown vector C is defined as: ⎫ ⎧ ur (r = a) /h1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ uθ (r = a) /h2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ uz (r = a) /h3 (3.108) C= ur (r = b) /h1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ uθ (r = b) /h2 ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ uz (r = b) /h3
The vector C contains the resultant modal stresses and displacements of the cylinder. The first three rows of the D matrix, which are composed of the displacement equations, are given as follows: D1,1 D1,2 D1,3
= αZn0 (αr) = αWn0 (αr) = αZn+1 (βr)
(3.109) (3.110) (3.111)
3. Vibration of Single-Layer Cylinder
D1,4 D1,5 D1,6
D2,1 D2,2 D2,3 D2,4 D2,5 D2,6
= αWn+1 (βr) n = Zn (βr) r n = Wn (βr) r
43
(3.112) (3.113) (3.114)
n = − Zn (αr) r n = − Wn (αr) r = ζZn+1 (βr)
(3.116)
= ζWn+1 (βr) = −βZn0 (βr) = −βWn0 (βr)
(3.118) (3.119) (3.120)
(3.115)
(3.117)
D3,1 D3,2
= −ζZn (αr) = −ζWn (αr)
D3,3
0 = −βZn+1 (βr) −
n+1 Zn+1 (βr) r
(3.123)
D3,4
0 = −βWn+1 (βr) −
n+1 Wn+1 (βr) r
(3.124)
D3,5 D3,6
= 0 = 0
(3.121) (3.122)
(3.125) (3.126)
Again, the appropriate functions to be used in place of W and Z are given in table 3.1. The last three rows of the D matrix are identical to the last three rows of the T matrix. The solution to equation (3.107), vector C, can yield displacements and stresses at any point within the cylinder.
3.10 Governing Equations for Free and Forced Vibrations Free vibration analysis requires free boundary stresses at the inner and outer radii of r = a and r = b. Thus, elements of vector S in equation (3.106) are zero. The nontrivial solution to the values of vector X in that equation requires that the determinant of the matrix T become zero. |T | = 0
(3.127)
44
3. Vibration of Single-Layer Cylinder
This is the frequency equation which has an infinite number of solutions for the frequency of ω for every given integer number for circumferential wave number n, and real value for the axial wave number ζ. After determining the nontrivial elements of vector X for any specific values of n and ζ and by substituting them into equation (3.107), modal displacements and stresses for any given radius can be determined. The governing equation for the forced vibration of thick cylindrical structure, excited by given harmonic boundary stresses, can be obtained by determining the vector X from equation (3.82) in terms of boundary stresses as presented in the following equation: X = T −1 S
(3.128)
The harmonic response of displacements and stresses on any point on a surface perpendicular to radial direction can be presented by substituting vector X from equation (3.128) into equation (3.107). C = DT −1 S
(3.129)
The frequency response for these displacements and stresses can be achieved by computing vector C for a given range of exciting frequencies.
3. Vibration of Single-Layer Cylinder
3.11 Key Symbols
Bn,x C D e E f (r) g1 , g2 , g3 gr (r) gθ (r) g3 (r) G, μ H,φ ˆır , ˆıθ , ˆız Jn , Yn n r r=a r=b S t T ur , uθ , uz u v1 v2 X z Wn , Zn In , Kn
Bessel Differential Operator vector of boundary displacements, 6 × 1 matrix of Bessel functions, 6 × 6 strain Young’s modulus solution of Bessel equation Bn,αx [f ] = 0 potential functions solution of Bessel equation Bn,βr [gr ] = 0 solution of Bessel equation Bn,βr [gθ ] = 0 solution of Bessel equation Bn,βr [g3 ] = 0 shear modulus, modulus of rigidity Lamé potential functions unit vectors along the axes r, θ, z Bessel functions integer number radial direction inner surface boundary outer surface boundary vector of boundary stresses, 6 × 1 time matrix of Bessel functions, 6 × 6 displacements in r, θ, z directions displacement vector velocity of propagation of the dilation wave velocity of propagation of the distortional wave vector of constants, vector of unknowns, 6 × 1 axial direction Bessel functions modified Bessel functions
α2 β2 δ ε θ λ μ, G ν ρ σ σ rr , σ θθ , σ zz
= ω 2 /v12 − ζ 2 = ω 2 /v22 − ζ 2 variation volumetric strain transverse direction Lame’s elastic constant shear modulus, modulus of rigidity Poisson’s ratio density normal stress direct stress in the r, θ, z directions
45
46
3. Vibration of Single-Layer Cylinder
τ τ rθ τ rz τ θz φ, H ω ζ
shear stress shear stress along θ and perpendicular to r shear stress along z and perpendicular to r shear stress along z and perpendicular to θ Lamé potential functions frequency axial wave number
0
differentiation of Bessel function with respect to its argument Laplacian operator gradient
∇2 ∇
4 Modal Analysis of Cylindrical Structures An overview, providing a physical understanding of the vibration modes for a single layer thick cylindrical structure, the main variables, and their general effect upon the response of the system, is given in this chapter. Furthermore, non-dimensional natural frequencies for a number of ratios of thickness to mean radius, different non-dimensional axial wavelengths, and circumferential wave numbers are tabulated and their respective modes are identified.
4.1 Frequency Factor To simplify the presentation of results and comparison with other established results for specific cases, natural and resonant frequencies are normalized and introduced as the frequency factor, which is defined by the following equation: ω Ω= (4.1) ωs where,
πv2 . (4.2) H ω s is defined as the lowest simple thickness shear frequency of an infinite plate of thickness H with the same elastic constants as those of the cylinder. The parameter ω is the excitation frequency, in units of rad/ s. For the purpose of comparing of results, the frequency factor is the same as the one defined by Armenakas et al. (1969). ωs =
4.2 Geometry of the Cylinder The definitions of the various geometric parameters are evident in figure 3.1. However, a more thorough description is given here. The variables a and b indicate the inner and outer radii of the cylinder, respectively. H is H.R. Hamidzadeh, R.N. Jazar, Vibrations of Thick Cylindrical Structures DOI 10.1007/978-0-387-75591-5_4, © Springer Science+Business Media, LLC 2010
48
4. Modal Analysis of Cylindrical Structures
defined as the thickness of the cylinder and R is the mean radius of the cylinder. In the numerical computation, a range of H/R = 0.0 to 2.0 was considered. The case of H/R = 2.0 approximates a solid cylinder. L is defined as the axial half wavelength.
4.3 Properties of the Medium The medium and dynamic behavior of single layer cylindrical structures depend on their elastic characteristics. These characteristics are redefined here to summarize needed equations. ρ is the density of the material. ν is defined as Poisson’s ratio. μ and λ are Lamé elastic constants which are defined as: μ=G (4.3) and λ=
2ν G. 1 − 2ν
(4.4)
For the homogeneous and isotropic materials, the shear modulus is related to the elastic modulus and the Poisson’s ratio as shown by the following relationship. E G= (4.5) 2 (1 − ν) Assuming that the material for the cylinder is a linear visco-elastic with a constant material loss-factor, both the elastic and shear moduli are presented as complex numbers, E = E 0 (1 + iη) G = G0 (1 + iη)
(4.6) (4.7)
where, G0 and E 0 are real parts of G and E respectively. η is defined as the loss-factor of the material.
4.4 Axial Wavelength The axial wavelength is dictated by the wavelength of the applied modal boundary stresses. Considering simply supported end conditions for the
4. Modal Analysis of Cylindrical Structures
49
L
z
FIGURE 4.1. Fundamental Axial Mode.
L
L
z
FIGURE 4.2. Second Axial Mode.
finite length cylinder case, the axial wave number, ζ, can be found with the following equation: π ζ= (4.8) L Recall that the variable L is defined as 1/2 of the axial wavelength. Figures 4.1 and 4.2 show the axial half wavelength for the fundamental and second longitudinal modes, respectively. For computation, the values of H/L were input from approximately 0.0 to 1.0. H/L = 0.0 corresponds to the infinite wavelength case. H/L > 0.0 corresponds to finite length cylinders with simply supported end conditions. Having the thickness of the cylinder and the length of L, the respective values of H/L and ζ are calculated for determination of the natural frequencies and resonant frequencies.
4.5 Modes of Vibration Depending on the value of nodal diameter (n), the axial wavelength (ξ), and specific values of f (r), gr (r), gθ (r), and g3 (r) in equations (3.16)-(3.19)
50
4. Modal Analysis of Cylindrical Structures
FIGURE 4.3. Breathing Mode n = 0.
infinite numbers of vibration modes can occur. In the following subsections, these modes are described.
4.5.1 Modes with Different Circumferential Wave Number These modes can be categorized in the following groups. 1. breathing mode for n = 0 (where any point on the cross section of the cylinder vibrates harmonically in its radial direction) 2. torsional mode for n = 0 (associated with transverse displacement of a long circumference of the cylinder) 3. axial mode for n = 0 (associated with axial vibration) 4. bending and axial shear modes for n = 1 (where the cross section of the cylinder remains the same) 5. lobar modes for n > 1 (where both radial and transverse displacements vary with sin(nθ)) By superposition, the displacement response of a single layer cylinder to an arbitrarily excited cylinder is composed of combinations of the infinite number of modes. Nevertheless, each mode can be excited by their respective modal boundary stresses. In this chapter, a number of individual mode shapes for cross sections of the cylinders are illustrated in Figures 4.3-4.8. The lower modes (n = 0, 1, 2, 3) have been found to be the dominant modes of vibration. In fact, analysis of the motion for a point in a cylinder can be approximated by summing the effects of the dominant modes. Therefore, addition of higher lobar modes yields a more accurate analysis of the cylinder’s response.
4. Modal Analysis of Cylindrical Structures
FIGURE 4.4. Bending and Axial Shear Modes n = 1.
FIGURE 4.5. First Lobar Mode, n = 2.
FIGURE 4.6. Second Lobar Mode, n = 3.
FIGURE 4.7. Third Lobar Mode, n = 4.
51
52
4. Modal Analysis of Cylindrical Structures
FIGURE 4.8. Fourth Lobar Mode, n = 5.
z uz ( z , t )
uθ ( z, t )
ur ( z, t )
FIGURE 4.9. Rigid Body Flexural Mode, n = 1.
The dominance of particular modes depends upon the type of excitation. Higher modes require exciting boundary stresses with very high frequencies. Therefore, these modes are usually not noticed in most cylindrical structures. The first four lobar modes, depicted in the above figures, indicate the oscillating patterns of the cross section of the cylindrical structure. These displacement patterns are shown by the dashed lines, while the nodes are indicated by small circles. The special cases of cylindrical vibrations include: the breathing mode n = 0, and the bending mode n = 1. The breathing mode requires radial displacement across the thickness of the cylinder in the absence of uθ and uz , with no nodal diameter. The bending mode is associated with no crosssectional distortion, while bending in the axial direction occurs. Figure 4.9 depicts the bending of a cylinder for this mode.
4.5.2 Thickness Modes Thickness modes are best introduced by using a mathematical explanation. Vibration of a cylinder is composed mathematically of an infinite number of circumferential wave numbers (n = 0, 1, 2, 3, . . .), i.e. the breathing, torsional, axial, and bending modes, and the lobar modes. Furthermore, for
4. Modal Analysis of Cylindrical Structures
Torsion
Axial
53
Breathing
FIGURE 4.10. Modes Associated with (n = 0) and One Node Circle.
FIGURE 4.11. Modes Associated with (n = 0) and Two Node Circle.
each circumferential number there is an infinite number of modes related to the infinite number of r values, where one of the displacement functions (ur , uθ , and uz ) vanishes. This is due to the nature of the involved Bessel functions. The thickness modes are occurring for each value of the circumferential wave number at specific values of ζ. Therefore, for the vibration of thick cylinders, there are infinite by infinite sets of vibration modes and for each mode, there are its natural frequency and its respective mode shape. Thus, whenever these types of structures are excited at a high natural frequency, a number of nodal circle(s) can occur. These nodal circle(s), describing the thickness modes, may be visualized by figures 4.10 and 4.11 that present two thickness modes for a vibrating cylindrical structure with a circumferential wave number of n = 0. In Figure 4.10, the first thickness mode has been presented; the dotted circle represents the thickness mode. Depending on the type of mode the associated displacement amplitude of every point on the node is zero. In other words, the dotted line indicates that on that position, no motion is observed. Furthermore, on the outside of the thickness circle nodes, the motion will be in the opposite direction to the points positioned inside the thickness circle nodes. In Figure 4.11, the two thickness modes are presented and the two thickness node circles are shown. The points in each of the three zones for the cross section of the cylinder are moving in alternate directions. In this spe-
54
4. Modal Analysis of Cylindrical Structures
cial case, the displacement directions could be (in, out, in) or oppositely (out, in, out). In these figures, the dashed lines indicate circular nodes.
4.5.3 Modes with Infinite Axial Wavelengths The infinite axial wavelength is denoted by specific cases where ζ = 0 (H/L = 0). This special case simplifies the stress matrix equation into two problems which can be independently solved. The independent solution of the two problems is commonly referred to as uncoupling of the solutions. As shown by Armenakas et al. (1969), the (r, z) plane motion is uncoupled from the longitudinal motion. By this uncoupling, the two problems that arise for infinite axial wavelengths are the plane strain vibration and axial (longitudinal) vibration. Plane strain vibration is denoted by radial and transverse motion, while uz = 0. The axial shear vibration happens by axial motion only. The vibration observed where ζ = 0 can be axial, plane strain, or a combination of both. For the special case, Breathing mode (n = 0) and (ζ = 0) , the transverse shear mode is uncoupled from the motion in the plane (r, z). Therefore, there are now three families of uncoupled solutions. This results in radial, transverse, or axial shear vibrations being observed. Also, the matrix solutions, for the plane strain and torsional modes, become identical. This results in the same natural frequencies for these modes.
4.5.4 Modes with Finite Axial Wavelengths When ζ has a value other than zero, finite length cylinders that are simply supported can also be analyzed to determine their natural frequencies, displacement, and stress mode shapes. In this analysis, both the Bessel functions and modified Bessel functions are utilized. For ζ 6= 0, the stress matrix cannot be uncoupled. Therefore, the plane strain vibration and the axial shear vibration are coupled. It should be noted that the nature of the breathing mode uncouples the transverse shear from the motion in the (r, z) plane.
4. Modal Analysis of Cylindrical Structures
z
55
ur ( z, t )
FIGURE 4.12. Radial Axisymmetric Mode.
z
uθ ( z, t )
ur ( z, t )
ur ( z, t )
uz ( z , t ) FIGURE 4.13. Axisymmetric Longitudinal and Breathing Modes.
4.5.5 Axially Symmetric Modes The axially symmetric vibration occurs for the special case of n = 0, the breathing mode. As shown by Armenakas et al. (1969), the stress matrix problem can be separated into two problems and then independently solved. The two problems that arise are those of longitudinal waves and torsional waves. Longitudinal waves involve only radial and axial displacements. Torsional waves involve nodal cylinders. Figure 4.12 depicts a radial axisymmetric mode and figure 4.13 represents the longitudinal mode. The torsional waves are more accurately described as vibration of the transverse sections of the cylinder. Each transverse section of the cylinder may rotate as a whole about the center, either clockwise or counterclockwise. The rotation direction depends upon the torsional mode. Figure 4.14 represents a typical torsional mode. Towest torsional mode is characterized by a continued rotation in a single direction (clockwise, for example). This rotation direction is continued through the length of the axial wave. The lowest torsional mode corresponds to β 2 = 0 which was presented in equation (3.37), and hence, ω = ζv2 .
(4.9)
Therefore, the torsional wave problem does not involve the Bessel equations that were presented earlier.
56
4. Modal Analysis of Cylindrical Structures
z uθ ( z, t )
uθ ( z, t )
ur ( z, t )
FIGURE 4.14. Axisymmetric Torsional Mode.
4.6 Forced and Free Modal Analysis The work of Fourier illustrates that any periodic function can be represented by a series of harmonic functions. In this chapter, harmonic excitation for boundary stresses are used to calculate resonance frequencies for coupled axial, torsional, flexural, and thickness modes. These harmonic excitations are for a specific vibration mode with certain values for axial and circumferential wave numbers, and excitation frequencies. With an appropriate harmonic exciting boundary stresses, the forced vibration frequency response for a given cylinder can be determined. In order to conduct modal analysis using forced vibration, constant values for amplitudes of inner boundary stresses can be assigned. σ rr τ rθ τ rz
= A1 cos (nθ) cos (ωt + ζz) = A2 sin (nθ) cos (ωt + ζz) = A3 cos (nθ) sin (ωt + ζz)
(4.10) (4.11) (4.12)
where: Ak is the amplitude of the stress on the inner boundary, n is the circumferential wave number, ω is the frequency of excitation, ζ is the axial wave number, (θ, z) are coordinate directions, and t is time.
To obtain a set of resonance frequencies for a particular axial wave length and a circumferential wave number, the matrix equation (3.129) can be used. Utilizing equations (4.10)-(4.12) for the inner boundary stresses and setting the outer boundary stresses to zero, numerical computation can be conducted to determine the stresses and displacements of the outer boundary of the vibrating cylinder for a wide range of frequencies and a small frequency interval. Then, the first six resonance frequencies for different circumferential wave numbers, n, can be computed by determining which
4. Modal Analysis of Cylindrical Structures
57
frequency factor causes the maximum absolute value of modal displacement. This is done independently for the first six modes of vibration for a specific cylindrical geometry. For each mode, different circumferential and axial wave numbers can be considered. The computed results for the outer boundary stresses should be zero. This may be used as a verification of the developed computer program for this analysis. It should be noted that the exciting stresses may be applied to the outer boundary with no expected difference in outcomes for the resonance frequencies. The inner or outer excited boundary stresses for the cylinder must be non-zero in all three directions to avoid missing a resonance mode. Without excitation in a certain direction, resulted amplitudes for a particular mode may be too small to be identified, especially for weak modes. To identify the resonance frequency factors, a frequency sweep can be made for the required parameters and the modal displacement data for each resonance frequency factor can be computed. To achieve higher accuracy for resonance frequency factors, groups of finer frequency factor range that include the resonant displacement, can be swept again. Thus, resonance frequency factors can be identified by determining the frequency factor that resulted in a greatest resonant displacement. Since some of the resonant frequencies cannot be identified using large frequency factor intervals, frequency sweeps should be conducted carefully. Small frequency factor intervals must be examined to assure an accurate resonance frequency factor. For example, a frequency factor interval of 0.1 × 10−5 may represent a frequency interval of 25 Hz, which is a large interval to identify the natural frequencies. An upper and lower triangular matrix method is recommended to compute the constants vector X in equation (3.106). The upper and lower triangular matrix method is more accurate than the matrix inversion method. For free vibration analysis, since boundaries are free from stresses using equation (3.129), one can compute the natural frequencies. Natural frequency factors can be determined by setting the determinant of the matrix DT −1 equal to zero. In this section, the inner or outer boundary of the cylinder must be stressed in all three of the defined directions to avoid missing a resonating mode. Without excitation in a certain direction, resultant amplitudes of a particular mode may be too small to be numerically identifiable. In this chapter, the first six natural frequency factors, Ω for the first five circumferential wave numbers and geometric parameters are computed and presented in a tabular format. The following values and range of parameters are considered for the computations: Poisson’s ratio: 0.3
58
4. Modal Analysis of Cylindrical Structures
Ratio of thickness to mean radius: 0.3 ≤ H/R ≤ 1.9 Ratio of thickness to axial half wavelengths: 0 ≤ H/L ≤ 1.0 Number of circumferential waves: n = 0, 1, 2, 3, 4, 5 Damping loss-factors: η = 0.0, 0.05, 0.1, 0.2, 0.3, 0.5, and 1.0 Normalized frequency factor: 0.003 ≤ Ω ≤ 6.0 (variable) The computed results of natural frequency factors for a number of modes and the above-mentioned parameters are presented. The results are tabulated, comparing the natural frequency factors of elastic cylinders to those obtained by Armenakas et al. (1969) in Tables 4.1-4.25. Furthermore, the results for the first six natural frequency factors are also provided for n = 5 in Tables 4.26-4.30.
In summary, the results are: Tables 4.1 through 4.25 present the first six frequency factors obtained by Armenakas et. al. (1969), and those computed using the presented technique. Each table offers data for four axial wavelengths.
Tables 4.1 to 4.5 include data for H/R = 0.3 and n = 0, 1, 2, 3, and 4 . Tables 4.6 to 4.10 provide data for H/R = 0.4 and n = 0, 1, 2, 3, and 4. Tables 4.11 to 4.15 present data for H/R = 0.5 and n = 0, 1, 2, 3, and 4. Tables 4.16 to 4.20 give data for H/R = 1.0 and n = 0, 1, 2, 3, and 4. Tables 4.21 to 4.25 contain data for H/R = 1.9 and n = 0, 1, 2, 3, and 4. Table 4.26 to 4.30 contain data for H/R = 0.3, 0.4, 0.5, 1.0, and 1.9, and n = 5.
Table 4.1 - Comparison of computed natural frequency factors with those of Ref [Armenakas et al.] for the first six modes (H/R = 0.3 and n = 0) H/L Resonant frequency factors, Ω 0.0 Present 0. 0.16301 1.00303 1.87872 2.00181 3.00116 0.1 0.5 1.0
Ref. Present Ref. Present Ref. Present Ref.
0. 0.13853 0.13829 0.31817 0.31790 0.74904 0.78474
0.16306 0.18950 0.18946 0.82804 0.82791 1.42011 1.42002
1.00344 1.02102 1.02055 1.33441 1.33441 1.91202 1.91156
1.87811 1.85702 1.85681 1.74405 1.74401 2.00302 2.00294
2.00174 2.03136 2.03137 2.33701 2.33665 2.82872 2.82864
3.00118 3.00069 3.00076 3.00064 3.00068 3.07883 3.07880
4. Modal Analysis of Cylindrical Structures
59
Table 4.2 - Comparison of computed natural frequency factors with those of Ref [Armenakas et al.] for the first six modes (H/R = 0.3 and n = 1) H/L Resonant frequency factors, Ω 0.0 Present 0. 0.09589 0.22528 1.00825 1.03407 1.85849 0.1 0.5 1.0
Ref. Present Ref. Present Ref. Present Ref.
0. 0.06139 0.06158 0.32103 0.32101 0.78863 0.78856
0.09584 0.16487 0.16484 0.51071 0.51065 1.00487 1.00483
0.22524 0.26031 0.26027 0.84139 0.84138 1.41669 1.41665
1.00819 1.01832 1.01826 1.13772 1.13767 1.43621 1.43618
1.03403 1.04542 1.04544 1.34524 1.34521 1.91693 1.91684
1.85845 1.84230 1.84229 1.74348 1.74344 2.00881 2.00872
Table 4.3 - Comparison of computed natural frequency factors with those of Ref [Armenakas et al.] for the first six modes (H/R = 0.3 and n = 2) H/L Resonant frequency factors, Ω 0.0 Present 0.03611 0.19171 0.35153 1.02246 1.08220 1.82141 0.1 0.5 1.0
Ref. Present Ref. Present Ref. Present Ref.
0.03596 0.05462 0.05456 0.33339 0.33332 0.80023 0.80015
0.19161 0.22790 0.22784 0.54012 0.54009 1.01920 1.01911
0.35149 0.38253 0.38247 0.87988 0.87975 1.41768 1.41769
1.00230 1.02959 1.02966 1.15074 1.15063 1.45432 1.45421
1.08213 1.09442 1.09432 1.37691 1.37683 1.93226 1.93221
1.82136 1.81199 1.81191 1.74263 1.74257 2.02644 2.02639
Table 4.4 - Comparison of computed natural frequency factors with those of Ref [Armenakas et al.] for the first six modes (H/R = 0.3 and n = 3) H/L Resonant frequency factors, Ω 0.0 Present 0.09559 0.28731 0.49311 1.04462 1.15526 1.78664 0.1 0.5 1.0
Ref. Present Ref. Present Ref. Present Ref.
0.09556 0.10642 0.10638 0.35972 0.35967 0.81961 0.81956
0.28721 0.30962 0.30951 0.58372 0.58365 1.04229 1.04226
0.49302 0.51786 0.51772 0.93781 0.93779 1.42546 1.42537
1.04453 1.05175 1.0517 1.17226 1.1722 1.47752 1.47741
1.15517 1.16745 1.16737 1.42714 1.42709 1.95653 1.95649
1.78652 1.78121 1.78116 1.74392 1.74383 2.05682 2.05677
60
4. Modal Analysis of Cylindrical Structures
Table 4.5 - Comparison of computed natural frequency factors with those of Ref [Armenakas et al.] for the first six modes (H/R = 0.3 and n = 4) H/L Resonant frequency factors, Ω 0.0 Present 0.17002 0.38264 0.63703 1.07711 1.24642 1.75996 0.1 0.5 1.0
Ref. Present Ref. Present Ref. Present Ref.
0.16997 0.17925 0.17917 0.40152 0.40149 0.84676 0.84673
0.38258 0.39831 0.39825 0.63771 0.6376 1.07351 1.07349
0.63697 0.65672 0.65666 1.00891 1.00885 1.43953 1.43945
1.07706 1.08285 1.08277 1.20242 1.20233 1.50501 1.50493
1.24635 1.25747 1.25739 1.49267 1.4926 1.98833 1.98827
1.75991 1.75723 1.75717 1.75105 1.75099 2.10052 2.10048
Table 4.6 - Comparison of computed natural frequency factors with those of Ref [Armenakas et al.] for the first six modes (H/R = 0.4 and n = 0) H/L Resonant frequency factors, Ω 0.0 Present 0. 0.21916 1.00618 1.88527 2.00314 3.00217 0.1 0.5 1.0
Ref. Present Ref. Present Ref. Present Ref.
0. 0.15291 0.15289 0.34113 0.34108 0.79281 0.79274
0.21915 0.22991 0.2298 0.83024 0.83018 1.42481 1.42474
1.00616 1.02342 1.02339 1.38841 1.33836 1.91519 1.91512
1.88521 1.86262 1.86255 1.74811 1.74805 2.00442 2.00431
2.00315 2.03353 2.0335 2.34974 2.33966 2.83123 2.83118
3.00212 3.00177 3.00171 3.00192 3.00181 3.08013 3.0801
Table 4.7 - Comparison of computed natural frequency factors with those of Ref [Armenakas et al.] for the first six modes (H/R = 0.4 and n = 1) H/L Resonant frequency factors, Ω 0.0 Present 0. 0.12817 0.29693 1.01857 1.06221 1.52004 0.1 0.5 1.0
Ref. Present Ref. Present Ref. Present Ref.
0. 0.05682 0.05695 0.34077 0.34071 0.79863 0.79858
0.12813 0.19202 0.19195 0.52102 0.52096 1.00901 1.00896
0.29686 0.32143 0.32138 0.85321 0.85315 1.41862 1.41855
1.0185 1.02542 1.02538 1.15314 1.15311 1.45363 1.45351
1.06217 1.07288 1.07276 1.35925 1.35922 1.92443 1.92435
1.52000 1.83817 1.83810 1.74697 1.74691 2.01521 2.01516
4. Modal Analysis of Cylindrical Structures
61
Table 4.8 - Comparison of computed natural frequency factors with those of Ref [Armenakas et al.] for the first six modes (H/R = 0.4 and n = 2) H/L Resonant frequency factors, Ω 0.0 Present 0.06216 0.25613 0.45802 1.04061 1.14901 1.80170 0.1 0.5 1.0
Ref. Present Ref. Present Ref. Present Ref.
0.06211 0.07467 0.07456 0.35448 0.35445 0.81671 0.81660
0.25593 0.28584 0.28573 0.57311 0.57309 1.03499 1.03494
0.45793 0.48176 0.48162 0.91569 0.91566 1.42068 1.42072
1.04057 1.04804 1.04801 1.17673 1.17671 1.48476 1.48478
1.14894 1.16051 1.16046 1.41891 1.41882 1.95017 1.95012
1.80159 1.79503 1.79501 1.74636 1.74632 2.04921 2.04913
Table 4.9 - Comparison of computed natural frequency factors with those of Ref [Armenakas et al.] for the first six modes (H/R = 0.4 and n = 3) H/L Resonant frequency factors, Ω 0.0 Present 0.14940 0.38311 0.63719 1.08285 1.27361 1.76325 0.1 0.5 1.0
Ref. Present Ref. Present Ref. Present Ref.
0.14935 0.16784 0.16774 0.39801 0.39794 0.84784 0.84781
0.38306 0.40070 0.40061 0.64377 0.64374 1.07577 1.07579
0.63714 0.65613 0.65601 1.00345 1.00342 1.43511 1.4351
1.08277 1.08863 1.08854 1.21633 1.2163 1.52371 1.52367
1.27357 1.28401 1.28391 1.50885 1.5088 1.98840 1.98837
1.76321 1.76050 1.76045 1.75444 1.75441 2.10849 2.10841
Table 4.10 - Comparison of computed natural frequency factors with those of Ref [Armenakas et al.] for the first six modes (H/R = 0.4 and n = 4) H/L Resonant Frequency Factors, Ω 0.0 Present 0.27422 0.50930 0.81564 1.13851 1.41924 1.74703 0.1 0.5 1.0
Ref. Present Ref. Present Ref. Present Ref.
0.27416 0.28137 0.28134 0.49699 0.49698 0.89261 0.89257
0.50922 0.52139 0.52136 0.72693 0.72697 1.12901 1.12897
0.81561 0.83026 0.83022 1.10202 1.102 1.46097 1.46091
1.13847 1.14421 1.14419 1.27141 1.27137 1.56882 1.56878
1.41926 1.42801 1.42795 1.61328 1.61323 2.03661 2.03657
1.74701 1.74687 1.74684 1.78673 1.78678 2.19128 2.19128
62
4. Modal Analysis of Cylindrical Structures
Table 4.11 - Comparison of computed natural frequency factors with those of Ref [Armenakas et al.] for the first six modes (H/R = 0.5 and n = 0) H/L Resonant Frequency Factors, Ω 0.0 Present 0. 0.2670 1.00971 1.89411 2.00503 3.00335 0.1 0.5 1.0
Ref. Present Ref. Present Ref. Present Ref.
0. 0.15702 0.15695 0.36921 0.36919 0.80320 0.80317
2.7673 0.28239 0.28235 0.83342 0.8334 1.43102 1.4310
1.00969 1.02704 1.02701 1.34367 1.34362 1.92001 1.91993
1.89408 1.87036 1.87032 1.75357 1.75354 2.00619 2.00612
2.00501 2.03641 2.03642 2.34739 2.34734 2.83461 2.83465
3.00336 3.00301 3.00297 3.00331 3.00328 3.08177 3.08180
Table 4.12 - Comparison of computed natural frequency factors with those of Ref [Armenakas et al.] for the first six modes (H/R = 0.5 and n = 1) H/L Resonant Frequency Factors, Ω 0.0 Present 0. 0.16070 0.36514 1.02384 1.10033 1.84605 0.1 0.5 1.0
Ref. Present Ref. Present Ref. Present Ref.
0. 0.0523 0.05213 0.35888 0.35884 0.81023 0.81027
0.16069 0.21866 0.21862 0.53635 0.53632 1.01476 1.01473
0.36513 0.38415 0.38412 0.86721 0.86723 1.42106 1.42103
1.02382 1.03447 1.03443 1.17273 1.17273 1.47601 1.47600
1.10034 1.11035 1.11037 1.37983 1.3798 1.93400 1.93396
1.84602 1.83411 1.83414 1.75152 1.75153 2.02428 2.02431
Table 4.13 - Comparison of computed natural frequency factors with those of Ref [Armenakas et al.] for the first six modes (H/R = 0.5 and n = 2) H/L Resonant Frequency Factors, Ω 0.0 Present 0.09377 0.32031 0.55468 1.06539 1.23762 1.78518 0.1 0.5 1.0
Ref. Present Ref. Present Ref. Present Ref.
0.09383 0.10292 0.10289 0.37305 0.37308 0.83409 0.83411
0.32034 0.34561 0.34556 0.61402 0.6140 1.05605 1.05603
0.55464 0.57411 0.57411 0.95429 0.95428 1.42503 1.42505
1.06533 1.07268 1.07271 1.21001 1.21002 1.52328 1.52322
1.23759 1.24787 1.24783 1.47873 1.47867 1.97016 1.97017
1.78511 1.78072 1.78065 1.75287 1.75288 2.08287 2.08290
4. Modal Analysis of Cylindrical Structures
63
Table 4.14 - Comparison of computed natural frequency factors with those of Ref [Armenakas et al.] for the first six modes (H/R = 0.5 and n = 3) H/L Resonant Frequency Factors, Ω 0.0 Present 0.23454 0.47801 0.76405 1.13182 1.42104 1.75286 0.1 0.5 1.0
Ref. Present Ref. Present Ref. Present Ref.
0.23251 0.23919 0.23916 0.44122 0.44121 0.87869 0.87864
0.47797 0.49257 0.49251 0.71208 0.71206 1.11808 1.11805
0.76409 0.77935 0.77933 1.06774 1.06769 1.44905 1.44901
1.13183 1.13805 1.13803 1.27300 1.27295 1.58022 1.58019
1.42106 1.42955 1.42952 1.61079 1.61076 2.02118 2.02114
1.75283 1.75207 1.75210 1.78285 1.78280 2.18172 2.18164
Table 4.15 - Comparison of computed natural frequency factors with those of Ref [Armenakas et al.] for the first six modes (H/R = 0.5 and n = 4) H/L Resonant Frequency Factors, Ω 0.0 Present 0.38839 0.63279 0.96802 1.22005 1.59996 1.78585 0.1 0.5 1.0
Ref. Present Ref. Present Ref. Present Ref.
0.38836 0.39401 0.39397 0.55009 0.55006 0.94455 0.94460
0.63276 0.64269 0.64279 0.82386 0.82384 1.19235 1.19236
0.96798 0.97918 0.97921 1.18787 1.18781 1.49093 1.49090
1.22007 1.22579 1.22574 1.35824 1.35817 1.64638 1.64634
1.59990 1.60522 1.60519 1.71428 1.71424 2.08588 2.08582
1.78581 1.78901 1.78891 1.89824 1.89820 2.29947 2.29949
Table 4.16 - Comparison of computed natural frequency factors with those of Ref [Armenakas et al.] for the first six modes (H/R = 1.0 and n = 0) H/L Resonant Frequency Factors, Ω 0.0 Present 0. 0.60063 1.04126 1.99844 2.02375 3.01637 0.1 0.5 1.0
Ref. Present Ref. Present Ref. Present Ref.
0. 0.16032 0.16029 0.55372 0.55369 0.89203 0.89200
0.60059 0.59785 0.59789 0.87516 0.87513 1.49805 1.49809
1.04127 1.06123 1.0612 1.39569 1.39566 1.97702 1.97696
1.99847 1.95261 1.95264 1.81700 1.81691 2.02383 2.02388
2.02371 2.07682 2.07688 2.39292 2.39297 2.87638 2.87635
3.01636 3.01613 3.01611 3.01849 3.01856 3.09891 3.09887
64
4. Modal Analysis of Cylindrical Structures
Table 4.17 - Comparison of computed natural frequency factors with those of Ref [Armenakas et al.] for the first six modes (H/R = 1.0 and n = 1) H/L Resonant Frequency Factors, Ω 0.0 Present 0. 0.32700 0.62737 1.11902 1.46298 1.83625 0.1 0.5 1.0
Ref. Present Ref. Present Ref. Present Ref.
0. 0.03567 0.03563 0.39268 0.39265 0.86593 0.86589
0.32698 0.36447 0.36452 0.67211 0.67212 1.07650 1.07651
0.62732 0.63778 0.63771 0.94887 0.94885 1.44301 1.44292
1.11903 1.12802 1.12806 1.30141 1.30142 1.65805 1.65803
1.46294 1.47032 1.47034 1.64581 1.64485 2.00044 2.00047
1.83627 1.83142 1.83143 1.79861 1.79854 2.14271 2.14177
Table 4.18 - Comparison of computed natural frequency factors with those of Ref [Armenakas et al.] for the first six modes (H/R = 1.0 and n = 2) H/L Resonant Frequency Factors, Ω 0.0 Present 0.31061 0.62227 0.89218 1.33056 1.72918 1.97099 0.1 0.5 1.0
Ref. Present Ref. Present Ref. Present Ref.
0.31064 0.31303 0.31299 0.47665 0.47661 0.91162 0.91153
0.62230 0.63767 0.63760 0.84182 0.84179 1.21529 1.21522
0.89214 0.90340 0.90339 1.13577 1.13570 1.48147 1.48146
1.33060 1.33591 1.33586 1.46501 1.46504 1.81264 1.81254
1.72920 1.73081 1.73082 1.78009 1.78015 2.06432 2.06424
1.97090 1.97507 1.97500 2.07508 2.07516 2.35507 2.35509
Table 4.19 - Comparison of computed natural frequency factors with those of Ref [Armenakas et al.] for the first six modes (H/R = 1.0 and n = 3) H/L Resonant Frequency Factors, Ω 0.0 Present 0.65844 0.88370 1.21732 1.61846 1.79892 2.35097 0.1 0.5 1.0
Ref. Present Ref. Present Ref. Present Ref.
0.65849 0.65951 0.65957 0.74093 0.74090 1.06027 1.06030
0.88365 0.89336 0.89336 1.05807 1.05812 1.37801 1.37805
1.21733 1.22517 1.22510 1.37811 1.37817 1.62387 1.62382
1.61842 1.62191 1.62188 1.72001 1.72004 2.01921 2.01918
1.79897 1.80255 1.80258 1.8806 1.8813 2.19132 2.19125
2.35095 2.34680 2.34683 2.37606 2.37614 2.55683 2.55677
4. Modal Analysis of Cylindrical Structures
65
Table 4.20 - Comparison of computed natural frequency factors with those of Ref [Armenakas et al.] for the first six modes (H/R = 1.0 and n = 4) H/L Resonant Frequency Factors, Ω 0.0 Present 0.95891 1.12625 1.55542 1.92250 2.00328 2.58983 0.1 0.5 1.0
Ref. Present Ref. Present Ref. Present Ref.
0.95897 0.95943 0.95948 1.00967 1.00961 1.24733 1.24734
1.12632 1.13382 1.13385 1.27070 1.27067 1.55409 1.55401
1.55537 1.56062 1.56058 1.66533 1.66535 1.86718 1.86716
1.92242 1.92501 1.92503 2.00609 2.00616 2.25758 2.25763
2.00330 2.00752 2.00757 2.09971 2.09966 2.38021 2.38029
2.58987 2.59121 2.59128 2.63101 2.63097 2.79232 2.79234
Table 4.21 - Comparison of computed natural frequency factors with those of Ref [Armenakas et al.] for the first six modes (H/R = 1.9 and n = 0) H/L Resonant Frequency Factors, Ω 0.0 Present 0. 1.19076 1.22962 2.18359 3.11852 3.17102 0.1 0.5 1.0
Ref. Present Ref. Present Ref. Present Ref.
0. 0.16048 0.16086 0.73574 0.73570 1.09661 1.09668
1.19078 1.15491 1.15497 1.15279 1.15285 1.73153 1.73159
1.22956 1.27831 1.27838 1.62351 1.62355 2.13091 2.13088
2.18355 2.18570 2.18575 2.24689 2.24687 2.52748 2.52757
3.11847 3.11781 3.11777 3.13850 3.13854 3.25918 3.25915
3.17090 3.17862 3.17854 3.32038 3.32032 3.66571 3.66578
Table 4.22 - Comparison of computed natural frequency factors with those of Ref [Armenakas et al.] for the first six modes (H/R = 1.9 and n = 1) H/L Resonant Frequency Factors, Ω 0.0 Present 0. 0.57201 0.84144 1.64887 1.99653 2.20241 0.1 0.5 1.0
Ref. Present Ref. Present Ref. Present Ref.
0. 0.02490 0.02482 0.37001 0.36991 0.87957 0.87963
0.57205 0.59572 0.59561 0.90074 0.90070 1.24771 1.24776
0.8415 0.88173 0.88167 1.09051 1.09049 1.54552 1.54558
1.64883 1.64911 1.64920 1.66998 1.68995 2.01172 2.01168
1.99648 2.00181 2.00173 2.09682 2.09668 2.29524 2.29528
2.20236 2.20791 2.20786 2.33612 2.33616 2.62975 2.62982
66
4. Modal Analysis of Cylindrical Structures
Table 4.23 - Comparison of computed natural frequency factors with those of Ref [Armenakas et al.] for the first six modes (H/R = 1.9 and n = 2) H/L Resonant Frequency Factors, Ω 0.0 Present 0.72557 0.94735 1.35982 2.07981 2.42737 2.93408 0.1 0.5 1.0
Ref. Present Ref. Present Ref. Present Ref.
0.72554 0.72182 0.72186 0.75746 0.75748 1.05153 1.05158
0.94727 0.96209 0.96213 1.17249 1.17258 1.48609 1.48616
1.35870 1.36564 1.36560 1.53017 1.53016 1.90947 1.90952
2.07987 2.08146 2.08141 2.12839 2.12836 2.36371 2.36363
2.42738 2.43034 2.43031 2.49581 2.49578 2.67252 2.67241
2.93410 2.93682 2.93680 2.99262 2.99255 3.13579 3.13575
Table 4.24 - Comparison of computed natural frequency factors with those of Ref [Armenakas et al.] for the first six modes (H/R = 1.9 and n = 3) H/L Resonant Frequency Factors, Ω 0.0 Present 1.12101 1.30305 1.86544 2.48597 2.87071 3.51898 0.1 0.5 1.0
Ref. Present Ref. Present Ref. Present Ref.
1.121040 1.11894 1.11891 1.13093 1.13098 1.31703 1.31705
1.30300 1.31229 1.31223 1.46463 1.46469 1.74033 1.74040
1.86541 1.87102 1.87098 1.99818 1.99814 2.30136 2.30141
2.48593 2.48778 2.48782 2.53759 2.53761 2.73571 2.73575
2.87063 2.87294 2.87287 2.92530 2.92534 3.08227 3.08235
3.51893 3.51701 3.51697 3.51529 3.51536 3.59928 3.59928
Table 4.25 - Comparison of computed natural frequency factors with those of Ref [Armenakas et al.] for the first six modes (H/R = 1.9 and n = 4) H/L Resonant Frequency Factors, Ω 0.0 Present 1.46231 1.64918 2.36764 2.87895 3.28911 3.93326 0.1 0.5 1.0
Ref. Present Ref. Present Ref. Present Ref.
1.46237 1.46269 1.46266 1.47591 1.47586 1.61059 1.61050
1.64924 1.65530 1.65521 1.76911 1.76909 2.00871 2.00866
2.36762 2.37159 2.37156 2.46815 2.46817 2.70408 2.70406
2.87892 2.88091 2.88085 2.92974 2.92969 3.10625 3.10630
3.28904 3.29106 3.29102 3.33792 3.33795 3.48378 3.48373
3.93327 3.93381 3.93377 3.94971 3.94973 4.02779 4.02776
4. Modal Analysis of Cylindrical Structures
H/L 0.0 0.1 0.5 1.0
H/L 0.0 0.1 0.5 1.0
H/L 0.0 0.1 0.5 1.0
H/L 0.0 0.1 0.5 1.0
H/L 0.0 0.1 0.5 1.0
67
Table 4.26 - Frequency factor for the fourth lobar mode (H/R = 0.3 and n = 5) Resonant Frequency Factors, Ω Present Present Present Present
0.25417 0.26239 0.45731 0.59445
0.47832 0.49354 0.69919 0.88231
0.77942 0.79532 1.08648 1.11230
1.11734 1.12208 1.24136 1.45931
1.35032 1.35948 1.56843 1.53638
1.74527 1.74486 1.76942 2.02629
Table 4.27 - Frequency factor for the fourth lobar mode (H/R = 0.4 and n = 5) Resonant Frequency Factors, Ω Present Present Present Present
0.39845 0.40439 0.56131 0.95431
0.63436 0.64342 0.81926 1.19153
0.98639 0.99721 1.20106 1.49728
1.20752 1.21343 1.34062 1.62047
1.56438 1.57142 1.70538 2.09436
1.77158 1.77327 1.87021 2.28832
Table 4.28 - Frequency factor for the fourth lobar mode (H/R = 0.5 and n = 5) Resonant Frequency Factors, Ω Present Present Present Present
0.55368 0.55521 0.68023 1.03348
0.78421 0.79153 0.94432 1.28447
1.15631 1.16317 1.30542 1.54826
1.32741 1.33211 1.45935 1.72306
1.69733 1.70053 1.79428 2.16430
1.95373 1.95942 2.08726 2.39241
Table 4.29 - Frequency factor for the fourth lobar mode (H/R = 1.0 and n = 5) Resonant Frequency Factors, Ω Present Present Present Present
1.20815 1.20908 1.24827 1.43739
1.36134 1.36727 1.48135 1.73420
1.90436 1.90722 1.98841 2.16539
2.21137 2.21417 2.28834 2.50414
2.30341 2.30722 2.38420 2.62235
2.87831 2.87902 2.91716 3.06425
Table 4.30 - Frequency factor for the fourth lobar mode (H/R = 1.9 and n = 5) Resonant Frequency Factors, Ω Present Present Present Present
1.78452 1.78471 1.80234 1.93018
2.11367 1.99432 2.08214 2.28934
2.85564 2.85932 2.93135 3.11183
3.25742 3.26486 3.31219 3.47362
3.70863 3.71034 3.75421 3.89223
4.31273 4.33920 4.35981 4.43732
68
4. Modal Analysis of Cylindrical Structures
Table 4.31 presents a comparison of computed natural frequency factors for the first three torsional modes (n = 0) for different values of H/L at H/R = 0.3 and their comparisons with those of Armenakas et al. (1963).
Table 4.31 - Comparison of computed torsional resonant frequency factors for different H/L with those of Ref [Armenakas et al.] (H/R = 0.3) H/L Resonant Frequency Factors, Ω 0.0 Present 0.0 1.0172 2.0088 0.1 0.2 0.4 0.6 0.8 1.0
Ref. Present Ref. Present Ref. Present Ref. Present Ref. Present Ref. Present Ref.
0.0 0.1000 0.1000 0.2000 0.2000 0.4000 0.4000 0.6000 0.6000 0.8000 0.8000 1.0000 1.0000
1.0172 1.0220 1.0220 1.0368 1.0270 1.0930 1.0930 1.1809 1.1810 1.2940 1.2940 1.4264 1.4264
2.0089 2.0112 2.0110 2.0188 2.0190 2.0480 2.0480 2.0964 2.0960 2.1622 2.1620 2.2438 2.2440
In addition, the effects of the material damping on the resonant frequency factors and modal loss factors associated with different circumferential wave numbers at different thickness to half wavelength ratios for visco-elastic thick cylinders are presented. The modal parameters for thick visco-elastic cylinders are achieved by using the radial, transverse, and axial displacement responses for forced vibrations when the cylinder is subjected to boundary stresses. These responses can be computed for any point on the medium. These frequency responses can be constructed in the form of Nyquist plots for a wide range of frequencies. Using an adequate number of data points around the vicinity of the resonance and assuming a single degree of freedom, the circle-fit method is employed to determine frequency, amplitude, and the modal loss factor at each resonance. The circle-fit method, which is described by Ewins (1978), maps the frequency response data points around a circle in the Nyquist diagram. It should be noted that the accuracy of these results will increase if the contributions of the nearby mode(s) does/do not change the response significantly for the data points near the resonance under consideration.
4. Modal Analysis of Cylindrical Structures
69
Table 4.32 presents the extracted resonant frequency factors for a cylinder with H/R = 0.3, for the circumferential wave numbers of n = 2, and for three different values of material loss factors. This table shows that, as expected, there is no significant difference in resonant frequency factors for different levels of material damping. Although, in a few cases, the resonant frequency factors have slightly changed, this is due to the fact that the responses for these resonances have no significant peaks.
Table 4.32 - Resonant frequency factors for different material loss factors (η) and thickness to half wavelength numbers (H/R = 0.3 and n = 2). Resonant Frequency Factors, Ω H/L η 1 2 3 4 5 0.01 0.0359 0.1919 0.3514 1.0222 1.0822 0.0 0.02 0.0359 0.1927 0.3514 1.0222 1.0820 0.05 0.0359 0.1963 0.3514 1.0216 1.0812 0.01 0.0499 0.2279 0.3825 1.0294 1.0949 0.1 0.02 0.0499 0.2283 0.3825 1.0282 1.0947 0.05 0.0501 0.2303 0.3825 1.0250 1.0925 0.01 0.1069 0.2933 0.4721 1.0463 1.1343 0.2 0.02 0.1069 0.2933 0.4721 1.0441 1.1341 0.05 0.1069 0.2945 0.4721 1.0453 1.1331 0.01 0.2513 0.4507 0.7323 1.1090 1.2810 0.4 0.02 0.2513 0.4507 0.7321 1.1078 1.2808 0.05 0.2569 0.4507 0.7321 1.1044 1.2796 0.01 0.4203 0.6329 1.0182 1.1989 1.4815 0.6 0.02 0.4219 0.6329 1.0177 1.1983 1.4815 0.05 0.4251 0.6341 1.0159 1.1947 1.4811 0.01 0.6061 0.8243 1.2548 1.3166 1.7051 0.8 0.02 0.6083 0.8245 1.2534 1.3158 1.7055 0.05 0.6107 0.8249 1.2558 1.3122 1.7079 0.01 0.7951 1.0192 1.4224 1.4543 1.9327 1.0 0.02 0.8031 1.0196 1.4224 1.4541 1.9345 0.05 0.8047 1.0184 1.4224 1.4527 1.9367
Tables 4.33 through 4.38 present modal loss factors for different values of material loss factors of η = 0.01, 0.02, and 0.05. The considered ratios of thickness to half wavelength: H/L = 0.0, 0.1, 0.2, 0.4, 0.6, 0.8, and 1.0, and the circumferential wave numbers: n = 0, 1, 2, 3, and 4 are presented. The special case of torsional vibration with n = 0 (axisymmetric) is presented in table 4.37. It should be noted that F indicates that data is not available, and FF indicates that data was not reliable due to the damped resonant peaks. The modal loss factors presented in these tables indicate that an
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increasing value of modal damping requires higher material loss factor. This relationship is evident in the presented tables.
Table 4.33 - Resonant frequency factors for different material loss factors (η) and thickness to half wavelength numbers (H/R = 0.3 and n = 2). Resonant Frequency Factors, Ω H/L η 1 2 3 4 5 0.01 F 1.634e-3 1.023e-2 1. 912e-2 1.939e-2 0.0 0.02 F 3.262e-3 2.045e-2 3.698e-2 2.002e-2 0.05 F 8.152e-3 4.687e-2 7.434e-2 9.920e-2 0.01 3.650e-4 2.254e-3 1.016e-2 1.910e-2 2.651e-2 0.1 0.02 3.483e-4 4.502e-3 2.042e-3 3.663e-2 4.043e-2 0.05 4.914e-4 1.124e-2 4.401e-2 FF 1. 003e-1 0.01 5.132e-3 2.988e-3 1. 002e-2 1. 972e-2 2.018e-2 0.2 0.02 1. 042e-2 5.944e-3 2.006e-2 3.766e-2 3.932e-2 0.05 2.972e-2 1.456e-2 7.1l8e-2 1. 084e-1 1.060e-1 0.01 1.1l8e-2 4.852e-3 1.014e-2 2.586e-2 1. 973e-2 0.4 0.02 2.230e-2 9.726e-3 2.045e-2 4.507e-2 4.100e-2 0.05 5.565e-2 2.463e-2 5.026e-2 4.796e-2 6.690e-2 0.01 1.34ge-2 6.924e-3 1.100e-2 2.427e-2 2.170e-2 0.6 0.02 2.683e-2 1.383e-2 2.196e-2 5.341e-2 4.251e-2 0.05 6.280e-2 3.476e-2 5.707e-2 1.078e-1 7.522e-2 0.01 2.912e-4 9.633e-3 1.241e-2 2.430e-2 2.343e-2 0.8 0.02 6.294e-3 1. 937e-2 2.576e-2 5.616e-2 4.326e-2 0.05 1.733e-3 5 .165e-2 5.972e-2 1.095e-1 4.461e-1 0.01 7.440e-3 1.455e-2 1.423e-2 2.361e-2 2.540e-2 1.0 0.02 1.01le-1 2.933e-2 2.924e-2 4.391e-2 4.345e-2 0.05 FF 6.913e-2 7.850e-2 5.322e-l 1.410e-1
4. Modal Analysis of Cylindrical Structures
71
Table 4.34 - Resonant frequency factors for different material loss factors (η) and thickness to half wavelength numbers (H/R = 0.3 and n = 2). Resonant Frequency Factors, Ω H/L η 1 2 3 4 5 0.01 F 9.737e-3 2.253e-3 1.046e-2 1.034e-2 0.0 0.02 F 1.583e-2 4.505e-3 2.113e-2 2.062e-2 0.05 F 3.073e-2 1.133e-2 8.550e-2 6.423e-2 0.01 1. 073e-2 4.776e-3 2.687e-3 1.060e-2 1.047e-2 0.1 0.02 2.413e-2 9.747e-3 5.360e-3 2.023e-2 2.141e-2 0.05 9.763e-2 2.795e-2 1.338e-2 1.918e-l 3.826e-2 0.01 1.422e-2 1.908e-3 3.351e-3 1.063e-2 1. 033e-2 0.2 0.02 2.927e-2 3.795e-3 6.703e-3 2 .163e-2 2.096e-2 0.05 2.390e-2 9.286e-3 1. 675e-2 FF 4.573e-2 0.01 1.623e-2 7 .353e-3 6.184e-3 1. 097e-2 1. 051e-2 0.4 0.02 3.268e-2 1.242e-2 1.232e-2 2.186e-2 2.045e-2 0.05 6.862e-2 7.306e-3 2.911e-2 5.478e-2 4.504e-2 0.01 1. 664e-2 8.661e-3 7.640e-3 1.185e-2 1.127e-2 0.6 0.02 3.274e-2 7.571e-4 1.548e-2 2.420e-2 2.236e-2 0.05 8.814e-2 1.405e-2 3.760e-2 5.800e-2 5.881e-2 0.01 1. 678e-2 1.218e-3 1.007e-2 1.295e-2 1.243e-2 0.8 0.02 3.458e-2 8.352e-3 2.013e-2 2.598e-2 2.530e-2 0.05 8.766e-2 3.476e-2 4.593e-2 6.321e-2 6.534e-2 0.01 5.081e-2 6.050e-3 6.837e-2 1.466e-2 1.472e-2 1.0 0.02 4.663e-2 1.693e-2 6.250e-2 3.826e-2 2.998e-2 0.05 8.718e-2 4.783e-2 FF 6.941e-2 9.415e-2
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Table 4.35 - Resonant frequency factors for different material loss factors (η) and thickness to half wavelength numbers (H/R = 0.3 and n = 2). Resonant Frequency Factors, Ω H/L η 1 2 3 4 5 0.01 3.853e-4 3.742e-3 3.527e-3 1.068e-2 1.083e-2 0.0 0.02 7.402e-4 7.107e-3 7.025e-3 2.138e-2 2.195e-2 0.05 1.807e-3 1.310e-2 1.753e-2 5.333e-2 5.231e-2 0.01 1.529e-1 1.208e-3 3.784e-3 1.057e-2 1.088e-2 0.1 0.02 1.330e-1 2.291e-3 7.576e-3 2.076e-2 2.147e-2 0.05 6.505e-2 4.193e-3 1.884e-2 5.314e-2 4.817e-2 0.01 6.557e-2 3.217e-4 4.418e-3 1.208e-2 1.095e-2 0.2 0.02 6.756e-2 6.072e-4 8.833e-3 2.197e-2 2.171e-2 0.05 7.841e-2 1.630e-3 2.231e-2 5.204e-2 5.525e-2 0.01 3.012e-2 4.602e-3 1.678e-2 1.116e-2 1.114e-2 0.4 0.02 6.312e-2 9.184e-3 2.693e-2 2.238e-2 2.262e-2 0.05 4.535e-l 2.229e-2 5.340e-2 6.284e-2 5.651e-2 0.01 2.624e-2 7.007e-3 1.527e-2 1.192e-2 1.185e-2 0.6 0.02 5.451e-2 1.398e-2 2.013e-2 2.317e-2 2.396e-2 0.05 1.671e-1 3.408e-2 6.542e-2 6.055e-2 6.113e-2 0.01 2.558e-2 8.985e-3 1.136e-2 1.308e-2 1.312e-2 0.8 0.02 5.009e-2 1.787e-2 2.216e-2 2.748e-2 2.635e-2 0.05 1.134e-1 5.079e-2 4.845e-2 6.662e-2 7.436e-2 0.01 4.775e-2 1.081e-2 1.802e-1 1.479e-2 1.510e-2 1.0 0.02 9.330e-2 2.163e-2 2.370e-l 2.912e-2 3.082e-2 0.05 7.209e-1 5.847e-2 5.673e-1 6.448e-2 7.357e-2
4. Modal Analysis of Cylindrical Structures
73
Table 4.36 - Resonant frequency factors for different material loss factors (η) and thickness to half wavelength numbers (H/R = 0.3 and n = 2). Resonant Frequency Factors, Ω H/L η 1 2 3 4 5 0.01 9.705e-4 1.236e-3 4.942e-3 1.092e-2 1.157e-2 0.0 0.02 1.919e-3 2 .343e-3 9.868e-3 2.183e-2 3.055e-2 0.05 4.772e-3 4.093e-3 2.487e-2 5.476e-2 5.930e-2 0.01 7.586e-3 2.166e-4 5.131e-3 1.092e-2 1.164e-2 0.1 0.02 6.719e-l 3 .975e-3 1. 025e-2 2.116e-2 2.291e-2 0.05 5.552e-2 2.107e-4 2.498e-2 5.509e-2 6.120e-2 0.01 2.863e-1 1. 152e-2 5.655e-3 1.103e-2 1.175e-2 0.2 0.02 5.285e-2 1.944e-2 1.132e-2 2.245e-2 3.131e-2 0.05 5.281e-l 3.835e-2 3.679e-2 5.343e-2 5.950e-2 0.01 8.292e-2 3.798e-3 7.161e-] 1.172e-2 1.199e-2 0.4 0.02 FF 7.588e-3 1.429e-2 2.419e-2 3.245e-2 0.05 3.052e-l 1.910e-2 3.559e-2 FF 5.931e-2 0.01 5.578e-2 6.374e-3 1.743e-2 1.199e•2 1.269e-2 0.6 0.02 1.353e-l 1.275e•2 3.161e-2 2.449e-2 2.637e-2 0.05 FF 3.092e-2 6.501e-2 5.935e-2 5.975e-2 0.01 4.383e-2 8.494e-] 1.345e-2 1.371e-2 1.378e-2 0.8 0.02 1.109e-l 2.245e-2 2.647e•2 2.794e-2 2.682e-2 0.05 1.730e-l 4.172e-2 FF 7.359e-2 7 .677e-2 0.01 3.532e-2 1.053e-2 FF 1.519e-2 1.531e-2 1.0 0.02 8.097e-2 2.076e-2 1.494e-l 2.986e-2 3.291e-2 0.05 1.275e-l 5.12]e-2 FF 7.349e-2 6.712e-2
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Table 4.37 - Resonant frequency factors for different material loss factors (η) and thickness to half wavelength numbers (H/R = 0.3 and n = 2). Resonant Frequency Factors, Ω H/L η 1 2 3 4 5 0.01 1.713e-3 F 6.374e-3 1. 124e-2 1.242e-2 0.0 0.02 3.406e-3 F 1.281e-2 2.231e-2 2.468e-2 0.05 8.518e-3 3.410e-3 3.233e-2 5.614e-2 6.348e-2 0.01 2.268e-1 FF 6.531e-2 1.116e-2 1.249e-2 0.1 0.02 5.434e-2 2.017e-3 1.302e-2 2.190e-2 2.533e-2 0.05 6.002e-2 5.733e-3 3.183e-2 5.753e-2 5.910e-2 0.01 4.979e-2 2 .075e-3 6.955e-3 1.120e-2 1.270e-2 0.2 0.02 3.120e-2 4.168e-3 1.388e-2 2.262e-2 2.521e-2 0.05 2.368e-1 1.085e-2 3.146e-2 6.742e-2 6.446e-2 0.01 1.130e-1 3.372e-3 8.369e-3 1. 880e-2 1.303e-2 0.4 0.02 6.256e-1 6.733e-3 1.669e-2 2.631e-2 2.565e-2 0.05 FF 1.702e-2 4.277e-2 3.682e-2 6.345e-2 0.01 1.244e-1 6.247e-3 3.395e-2 1.232e-2 1.363e-2 0.6 0.02 5.707e-1 1.245e-2 3.522e-2 2.545e-2 2.845e-2 0.05 FF 3.273e-2 5.614e-2 7.845e-2 6.227e-2 0.01 7.477e-2 8.407e-3 1.478e-2 1.501e-2 1.458e-2 0.8 0.02 1.498e-1 1.718e-2 3.342e-2 2.931e-2 2.915e-2 0.05 2.540e-1 4.192e-2 9.604e-2 8.648e-2 8.705e-2 0.01 8.008e-2 1.044e-2 2.763e-2 1.509e-2 1.636e-2 1.0 0.02 1.072e-1 2.097e-2 7.136e-2 3.024e-2 3.421e-2 0.05 1.599e-1 5.289e-2 2.083e-1 6.722e-2 7.504e-2
4. Modal Analysis of Cylindrical Structures
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Table 4.38 - Modal loss factors for different material loss factors (η) and thickness to half wavelength numbers (torsional mode n = 0). (H/R = 0.3) H/L 0.0
0.1
0.2
0.4
0.6
0.8
1.0
η 0.01 0.02 0.05 0.01 0.02 0.05 0.01 0.02 0.05 0.01 0.02 0.05 0.01 0.02 0.05 0.01 0.02 0.05 0.01 0.02 0.05
Modal loss factor 2 3 F 1.019e-2 2.045e-2 F 2.084e-2 4.022e-2 F 4.978e-2 9.826e-2 FF 1. 024e-2 2.038e-2 FF 2.073e-2 4.037e-2 FF 4.994e-2 7.059e-2 8.290e-3 1. 038e-2 2.042e-2 1.14le-2 1.037e-2 2.0l9e-2 FF 5.318e-2 8.694e-2 7.764e-3 1. 096e-2 1.96le-2 9.278e-3 2.2l5e-2 4.ll2e-2 FF 5.287e-2 7.432e-2 6.739e-3 1. 183e-2 2.109e-2 2.ll7e-4 2.465e-2 4.222e-2 5.034e-4 5.62le-2 6.938e-2 1. 809e-4 1.293e-2 2.135e-2 3.740e-4 2.645e-2 4.277e-2 4.24le-2 7.43le-2 6.724e-2 2.956e-4 1. 427e-2 2.290e-2 5.599e-4 3.759e-2 4.40le-2 4.83le-2 7.176e-2 7.694e-2 1
Table 4.39 presents the first natural frequency factors for axial bending, associated with different circumferential numbers, for two different simply supported thin elastic cylindrical shells. This table depicts a comparison of the presented solution and those extracted from Armenakas et al. (1969) and Markus, (1988), as well as, the reported results using the Flugge’s thin shell theory (see Markus 1988). The non-dimensionalized frequency factor utilized by Markus (1988) is of a different form than the frequency factor introduced in this text and it is presented as: r ρ 0 Ω = Rω (4.13) (1 − ν 2 ) E To convert the frequency factor used by Markus (1988), Ω0 , to the one
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4. Modal Analysis of Cylindrical Structures
introduced in this text the following equation can be used. Ω = 0.538041909Ω0
H R
(4.14)
For the results presented in Table 4.31 the frequency factor of Ω0 is used. These results are for two cases of thin cylindrical shells with H/R = 0.05 and H/R = 0.02 in combination with R/L = 0.05 (long shell). Markus (1988) inferred the following four conclusions from the comparison with those of Flugge’s:
1. Shell theory always produces values of Ω0 larger than those based on theory of elasticity.
2. The observed relative error never exceeded 10% and therefore differences in the values obtained in the two theories are negligible for very thin cylinders.
3. The approximate theories are satisfactory for thin cylinders from the engineering viewpoint.
4. Flugge’s results fall within 0.01% of all other investigators using other versions of the shell theory.
As shown, the results reported by the present method and those of Markus (1988) are in relatively similar. In particular, the results presented by Armenakas et al. (1969) are consistently higher than those of Markus. It should be noted that results listed under Armenakas et al. for the long cylinder have been linearly interpolated from their publication. Obviously, one should expect an appreciable error by using such a technique to interpolate data, mainly for lower frequency factors. However, for higher modes the error is reduced. It should be noted that Armenakas et al. (1969) did not provide results for the extremely thin cylinder.
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77
Table 4.39 - Comparison of Fundamental Frequency Parameters for a thin long cylinder R/L = 0.05. Resonant Frequency Factors, Ω H/R n Flugge Armenakas et al. Markus Present 0 0.096191 No Data 0.092930 0.092930 1 0.016352 No Data 0.016101 0.016101 0.002 2 0.005529 No Data 0.005452 0.005446 3 0.005060 No Data 0.005037 0.005046 4 0.008538 No Data 0.008534 0.008531 0 0.096191 0.149198 0.092929 0.092929 1 0.016349 0.023820 0.016106 0.016095 0.05 2 0.392916 0.040889 0.039233 0.039235 3 0.109782 0.109842 0.109477 0.109482 4 0.210197 0.209091 0.209008 0.209007
4.7 Remarks In general, the natural frequency factors presented in this chapter for thick circular cylinders are very similar with those given by Armenakas et al. (1969). Nevertheless, in some cases, Armenakas’ results slightly differ from the presented computed results. These deviations occur mostly in the fifth digit after the decimal point for the computed frequency factors and are believed to be caused by the inaccuracy of the techniques used for the constructions of Bessel and modified Bessel functions. Furthermore, it should be noted that due to the nature of Bessel functions of the second kind, for very low frequency factors and very short wavelength ratios, they are not consistently accurate in these conditions.
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4.8 Key Symbols A E E0 F G, μ G0 H L n n=0 n=1 n>1 r r=a r=b R t ur , uθ , uz X z
stress amplitude Young’s modulus real parts of E direct or shear stresses shear modulus, modulus of rigidity real parts of G thickness of the cylinder axial half wavelength wave number breathing mode rigid body mode lobar modes radial direction inner surface boundary outer surface boundary mean radius of the cylinder time displacements in r, θ, z directions vector of constants, vector of unknowns axial direction
η θ λ μ, G ν ρ σ σ rr , σ θθ , σ zz τ τ rθ τ rz τ θz ω ωs Ω ζ = π/L
loss-factor transverse direction Lame’s elastic constant shear modulus, modulus of rigidity Poisson’s ratio density normal stress direct stress in the r, θ, z directions shear stress shear stress along θ and perpendicular to r shear stress along z and perpendicular to r shear stress along z and perpendicular to θ excitation frequency the lowest simple thickness shear frequency frequency factor axial wave number
5 Vibration of Multi-Layer thick cylinders Free vibration of a laminated multi-layer, thick infinitely long cylindrical structure is presented by adopting the analytical solution for a single-layer cylinder. The solution is achieved by determining the displacements and stresses for each interface and by complying with requirements at the interfaces. A propagator matrix relating the boundary displacements to boundary stresses is developed. Dimensionless natural frequencies and modal loss factors for different circumferential and axial wave numbers are determined. The validity of the proposed method is verified by comparing the results for one-, two-, and three-layer elastic cylinders with properties similar to those reported for an equivalent single layer.
5.1 Introduction Laminated cylinders have widespread use in engineering. Pipe systems, aircraft, submarines, missiles, rockets, and power transmission shafts are typical applications for this type of cylindrical structure. When the frequency of an acting force on a cylinder coincides with the natural frequency of the structure, the phenomenon of resonance occurs. Resonance is associated with large displacements and stresses, and may cause failure in the cylinder. Therefore, determining the natural frequency is vital and has a direct impact on the design of composite cylindrical structures. To control large displacements and stresses, damping treatment using visco-elastic material with a complex shear modulus are employed. In constrained layer damping treatment, a visco-elastic material is sandwiched between two elastic layers. The visco-elastic layer dissipates much of the energy within the visco-elastic cylinder in the form of heat during shear deformation, when subjected to harmonic vibration. In this configuration, the middle layer allows shear displacement, which converts dissipation of energy into heat, while the elastic layers give the cylindrical structure rigidity and stability. Many researchers have investigated the vibration of cylinders using different analytical approaches, such as: shell theory, curved beam theory, and H.R. Hamidzadeh, R.N. Jazar, Vibrations of Thick Cylindrical Structures DOI 10.1007/978-0-387-75591-5_5, © Springer Science+Business Media, LLC 2010
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5. Vibration of Multi-Layer thick cylinders
theory of elasticity. Most reported studies of the vibrations in constrained layer damped cylinders are based on the assumptions for thin shell cylinders. The assumptions used in shell theory for cylindrical structures may not always be valid. In fact, a significant number of cylindrical structures are considered to be thick. In spite of this, few studies have been directed to the analysis of free vibrations of thick cylindrical structures.
5.2 Historical Background In the foregoing section, a summary of notable research work on this topic is presented. Initial research on vibrations of sandwich cylindrical shells was conducted by White (1960) and Yu (1960), who considered visco-elastic properties for three-layered cylindrical structures. Jones and Salerno (1965) investigated the damping effect in the forced axisymmetrical vibration of a cylindrical sandwich shell. The sandwich cylinders considered by Yu (1960), Jones and Salerno (1965) were infinitely long or simply supported. Pan (1968) studied axisymmetrical vibrations of a circular sandwich shell using a visco-elastic layer with a complex shear modulus. He derived and presented the partial differential equations for the motion. The solution of these differential equations yields the frequency equation and the corresponding modal loss-factor after satisfying the boundary conditions. Lu et al. (1973) introduced damping material in the middle layer and analyzed the mechanical impedance of damped three-layered sandwich rings by using the approximate method. They considered the effects of the operational temperature and frequency ranges on the visco-elastic material. The experimental data compared reasonably well with their theoretical predictions for lower modes. Hamidzadeh and Chandler (1991) analytically simulated one of the cases presented by Lu et al. (1973) using forced vibration analysis. Their results compared satisfactorily with those of Lu et al. (1973).
Hamidzadeh (1990) presented solutions for lobar vibration of three layered sandwiched cylinders of infinite extend. Hamidzadeh and Chandler (1991) also reported a solution to circumferential vibrations of three-layered, sandwiched, thick cylinders using the plane strain assumptions. Rattanawangcharoen and Shah (1990) studied wave propagation in a laminated isotropic cylinder. The dispersion relation for a multi-layered elastic cylinder was evaluated by a propagator matrix and non-dimensional natural frequencies for a two-layered cylinder for a special circumferential wave number, n = 1 , and a range of axial wave numbers were calculated.
5. Vibration of Multi-Layer thick cylinders
81
Hawkes and Soldatos (1992) studied vibrations of multi-layered laminated cylinders. They achieved the solution using a method of successive approximations. In their approach, the exact governing equations for the cylinders were replaced by a set of approximate equations that were solved analytically. They compared their results with those of Armenakas et al. (1969).
Hamidzadeh and Sawaya (1995) presented results of their analysis of constrained layer damping of thick cylinders. They developed a propagator matrix relating displacements and stresses at the interface of a layer to its adjacent layer. They introduced their analytical solution for determining the natural frequencies and modal loss-factors for a number modes of vibrations for the three-layer, thick, sandwiched cylinders of infinite length. Hamidzadeh and Jiang (1995) used the same analytical method and presented modal loss-factors for three-layered sandwiched thick cylinders in a number of case studies. Hamidzadeh (2009) adopted the same analytical method and considered the effect of visco-elastic core thickness on modal loss-factor of a thick three-layer cylinder.
5.3 Scope of the Chapter The governing equation for thick, axially symmetric, as well as non-symmetric isotropic visco-elastic multi-layered cylinders of infinite extent will be presented. The three-dimensional analytical solution developed in chapter three using wave propagation, which applies equally to thick or thin cylinders, will be applied. The method presented here does not use any approximation. It relates the modal displacements and stresses of the first layer to the last layer in the cylinder using a propagator matrix. The advantage of this method is that it can deal with any number of layers. The method is discussed in detail in this chapter. As reported by Hamidzadeh and Sawaya (1995), the solution is achieved by determining the displacements and stresses for each interface and by complying with the compatibility requirements at the boundaries. A propagator matrix relating boundary displacements and stresses between the inner and outer layers is developed. This method can consider any number of elastic and visco-elastic layers for a cylinder. Non-dimensional natural frequencies and modal loss-factors for different circumferential and axial wave numbers are calculated for both axial and nonaxisymmetic modes. Case studies and the required computation were conducted to determine the natural frequencies for single-layer, two-layer,
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5. Vibration of Multi-Layer thick cylinders
and three-layer cylinders. Results of these case studies are compared with those reported by Armenakas et al. (1969) for elastic cylinders with similar geometries. This was done to verify the accuracy of the analytical method relating the inner layer to the outer one (Hamidzadeh and Sawaya, 1995). The analytical solutions presented in chapter 3 are used to develop the analytical solutions for determination of modal characteristics for fully covered, three-layer, and multi-layer cylindrical structures.
5.4 Modal Displacements and Stresses As was presented in chapter three, using equations (3.108) and (3.84), modal displacements and stresses can be written in terms of Bessel functions in the following equation: ⎫ ⎡ ⎫ ⎧ ⎤⎧ D1,1 D1,2 D1,3 D1,4 D1,5 D1,6 ⎪ A1 ⎪ ur /h1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ D2,1 D2,2 D2,3 D2,4 D2,5 D2,6 ⎥ ⎪ ⎪ uθ /h2 ⎪ B1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎥ ⎨ ⎬ ⎢ ⎬ ⎨ ⎥ uz /h3 D D D D D D A 3,1 3,2 3,3 3,4 3,5 3,6 ⎥ 2 ⎢ =⎢ ⎥ σ rr /h1 ⎪ B2 ⎪ ⎪ ⎪ ⎪ ⎢ T1,1 T1,2 T1,3 T1,4 T1,5 T1,6 ⎥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ ⎦ τ /h T T T T T T A3 ⎪ ⎪ ⎪ ⎪ ⎪ rθ 2 ⎪ 2,1 2,2 2,3 2,4 2,5 2,6 ⎪ ⎪ ⎪ ⎩ ⎭ ⎭ ⎩ τ rz /h3 T3,1 T3,2 T3,3 T3,4 T3,5 T3,6 B3 (5.1) where elements of the matrix are in terms of the Bessel functions and were presented in chapter 3. It should be noted that: ur is the displacement in the r direction, uθ is the displacement in the θ direction, uz is the displacement in the z direction, σrr is the normal radial stress in the r direction, τ rθ is the shear stress in the θ direction and normal to r, τ rz is the shear stress in the z direction and normal to r, and X is the constant vector given previously in equation (3.83) Then equation (5.1) can be presented as: {d}n = [D]n {X}n
(5.2)
where, the vector {d}n is: {d}n =
n ur h1
uθ h2
uz h3
σ rr h1
σ rθ h2
σ rz oT h3 n
(5.3)
and [D]n previously appeared in equation (3.107), and n is the circumferential wave number.
5. Vibration of Multi-Layer thick cylinders
83
Interface II Interface I
Layer III Layer II Layer I
r0
6 4 5 2 3 1
r1 z r2 r3
FIGURE 5.1. Cross Section of a Three Layered Cylinder.
5.5 Propagator Matrix for a Three Layer Cylinder The layered cylinder is assumed to be formed by three different visco-elastic layers bounded by inner and outer interfaces (see Figure 5.1). The third layer has a thickness of H3 = r3 − r2 and is bounded by inner and outer interfaces located at radii of r2 and r3 , respectively. Each of the viscoelastic cylinders is characterized by a complex compressional wave velocity v1 , complex shear wave velocity v2 , density ρ, Poisson’s ratio ν, and a material loss-factor η. The mathematical formulation is used to develop a relation between displacements and stresses at the boundaries of each layer and consequently obtains a propagator matrix relating displacements and stresses of the inner boundaries to the outer ones. To build the propagator matrix for a three layered cylinder, the relations for stresses and displacements at any location in the cylindrical structure, defined by equation (5.1) is used. In Figure (5.1), points (1) and (2), (3) and (4), and (5) and (6), lay on layer I, layer II and layer III, respectively. All points laying on the same layer comprise the same unknown vector {X}n . Points (2) and (3), and (4) and (5), lay on interface I and II, respectively. Therefore, points (2) and (3), and (4) and (5), have the same displacements and stresses. As presented in equation (5.2), the displacements and stresses vector for any point on the cylinder can be given as follows: {d}np = [D]np {X}np
(5.4)
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5. Vibration of Multi-Layer thick cylinders
where, n is the circumferential wave number and p is the point number. Using the general equation of (5.1), the following equations can be written for each point in the medium. Writing the displacements and stresses vector for the six points presented in Figure 5.1 and dropping subscript n for simplification one can write the following equations for points (1) through (6). point (1): {d}1 = [D]1 {X}1
(5.5)
{d}2 = [D]2 {X}2
(5.6)
{d}3 = [D]3 {X}3
(5.7)
{d}4 = [D]4 {X}4
(5.8)
{d}5 = [D]5 {X}5
(5.9)
{d}6 = [D]6 {X}6
(5.10)
point (2): point (3): point (4): point (5): point (6):
Points (1) and (2) are on the same layer, therefore: {X}1 = {X}2
(5.11)
Therefore, using these equalities and equations (5.5) and (5.6) yield: {d}2 = [D]2 [D]−1 1 {d}1
(5.12)
Points (3) and (4), and (5) and (6), are on layers two and three respectively, this implies that points (3) and (4), and (5) and (6), have the same unknown constants, therefore: {X}3 {X}5
= {X}4 = {X}6
(5.13) (5.14)
Using the same procedure used for the first layer which resulted the equation (5.12), one can write the following two equations for the second and the third layers. {d}4
{d}6
−1
= [D]4 [D]3 {d}3 =
[D]6 [D]−1 5
{d}5
(5.15) (5.16)
5. Vibration of Multi-Layer thick cylinders
85
Points (2) and (3) lay on different layers but they have the same displacement and stresses. Similarly, points (4) and (5) are on different layers but have the same displacement and stresses. This implies that: {d}2 = {d}3
{d}4 = {d}5
(5.17)
Substituting {d}3 for {d}2 in (5.12), it yields: −1
{d}3 = [D]2 [D]1 {d}1
(5.18)
{d}5 = [D]4 [D]−1 3 {d}3
(5.19)
Similarly, one can write:
Introducing [PI ], [PII ], and [PIII ] according to the following expressions: [PI ] = [D]2 [D]−1 1
(5.20)
−1 [D]4 [D]3 −1 [D]6 [D]5
(5.21)
[PII ] = [PIII ] =
(5.22)
Combining equations (5.16), (5.18), and then (5.19), the relationship between point (1) and point (6) is established and is equal to: {d}6 = [PIII ] [PII ] [PI ] {d}1
(5.23)
[P ] = [PIII ] [PII ] [PI ]
(5.24)
{d}6 = [P ] {d}1
(5.25)
Introducing [P ] as: Therefore, For free vibrations, the stresses at the inner and outer radii are equal to zero. Therefore equation (5.25) can be written as: ⎧ ur ⎫ ⎧ ur ⎫ ⎪ ⎪ ⎤⎪ ⎡ ⎪ h ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ h1 ⎪ P1,1 P1,2 P1,3 P1,4 P1,5 P1,6 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ u uθ1 ⎪ θ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ P2,1 P2,2 P2,3 P2,4 P2,5 P2,6 ⎥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥ ⎢ h2 ⎬ h ⎨ ⎬ ⎨ u 2 ⎢ P3,1 P3,2 P3,3 P3,4 P3,5 P3,6 ⎥ u z z ⎥ ⎢ =⎢ (5.26) ⎥ ⎪ h3 ⎪ h3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ P4,1 P4,2 P4,3 P4,4 P4,5 P4,6 ⎥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ P5,1 P5,2 P5,3 P5,4 P5,5 P5,6 ⎦ ⎪ 0 ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 0 P P P P P P ⎪ ⎪ ⎪ ⎪ 6,1 6,2 6,3 6,4 6,5 6,6 ⎩ ⎭ ⎭ ⎩ 0 0 6 1 The above equation yields to: ⎡
P4,1 ⎣ P5,1 P6,1
P4,2 P5,2 P6,2
P4,3 P5,3 P6,3
⎧ ⎤⎪ ⎪ ⎪ ⎨ ⎦ ⎪ ⎪ ⎪ ⎩
ur h1 uθ h uz2 h3
⎫ ⎪ ⎪ ⎪ ⎬
⎧ ⎫ ⎨ 0 ⎬ 0 = ⎪ ⎩ ⎭ ⎪ 0 ⎪ ⎭ 1
(5.27)
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5. Vibration of Multi-Layer thick cylinders
For the nontrivial solutions of the above equation, the determinant of the 3 × 3 matrix must be equal to zero and the roots of the determinant are the natural frequency of the composite cylinder. In other words, the determinant of the coefficient matrix in equation (5.27) provides the frequency equation for the system. It should be indicated that elements of the above matrix are all related to the circumferential and axial wave numbers, the material loss-factor, and the geometry of each layer. Dimensionless natural frequencies and modal loss-factors for different modes can be computed using Muller’s iterative method as described by Conte (1965).
5.6 Propagator Matrix for a Multi-Layered Thick Cylinder In summary, a multi-layered composite cylinder for a set of a specific wave number can be formulated in the manner described in the previous section. Considering Figure 5.2, which presents a multi-layered composite cylinder for the ith and (i + l)th interface, we can write: {d}i {d}i+1
= [D]i {X}i = [D]i+1 {X}i+1
(5.28) (5.29)
Since for the ith layer {X}i is the same as {X}i+1 , one can write the following equation: {d}i+1 = [D]i+1 [D]−1 (5.30) i {d}i or where,
{d}i+1 = [P ]i+1 {d}i
(5.31)
[P ]i+1 = [D]i+1 [D]−1 i .
(5.32)
Writing equation (5.31) for all layers and eliminating vector {d} for all intermediate interfaces results in a relation between displacements and stresses vectors at the inner and outer boundaries of the sandwiched cylinder. {d}i+1 = [P ]i+1 [P ]i [P ]i−1 · · · [P ]1 {d}1 (5.33) It can be rewrite as {d}i+1 = [P ] {d}1
(5.34)
[P ] = [P ]i+1 [P ]i [P ]i−1 · · · [P ]1 .
(5.35)
where,
5. Vibration of Multi-Layer thick cylinders
Hi+1
87
ith layer
r
H θ
r0
ri+1 ri
FIGURE 5.2. Cross Section of a Multi-Layered Circular Cylinder.
The 6 × 6 matrix [P ] is the propagator matrix. For free vibration, as was presented for the three layered cylinder equation (5.34) reduces to: ⎡
P4,1 ⎣ P5,1 P6,1
P4,2 P5,2 P6,2
P4,3 P5,3 P6,3
⎧ ⎤⎪ ⎪ ⎪ ⎨ ⎦ ⎪ ⎪ ⎪ ⎩
ur h1 uθ h uz2 h3
⎫ ⎪ ⎪ ⎪ ⎬
⎧ ⎫ ⎨ 0 ⎬ 0 = ⎪ ⎩ ⎭ ⎪ 0 ⎪ ⎭
(5.36)
1
5.7 Natural Frequencies and Modal Loss-Factors The nontrivial solutions of equation (5.27) for three-layer cylinders or (5.36) for multi-layer cylinders requires that: ¯ ¯ ¯ P4,1 P4,2 P4,3 ¯ ¯ ¯ ¯ P5,1 P5,2 P5,3 ¯ = 0 (5.37) ¯ ¯ ¯ P6,1 P6,2 P6,3 ¯
This equation provides the natural frequencies of the composite sandwich cylinders with infinte extend. Since elements of the matrix are in terms of Bessel functions, the equation (5.39) has an infinite number of solutions for any circumferential wave number. In this text, an attempt is made to determine only the first six roots of equation (5.39). As mentioned in chapter four, Muller’s method which is an extension of the secant method is used. This method may be used to find any prescribed number of roots,
88
5. Vibration of Multi-Layer thick cylinders
real or complex, of an arbitrary function. Muller’s method computes both real and complex roots without requiring any derivative of the function. A detailed description of Muller’s method is presented by Conte (1965), Beckett and Hurt (1967), Conte and Boor (1980). The roots of equation (5.37) are natural frequencies of the layered cylinders and are in complex form for visco-elastic materials: ω =
